CURRICULUM VITAE and STUDIORUM

FERNANDA PAMBIANCO

PERSONALBorn: December 25th , 1964, Perugia, Italy.

EDUCATION, DEGREESStudied Mathematics at University of Perugia (Perugia, Italy) 1983-1987.

Degrees: Degree in Mathematics, taken summa cum laude, at University of Perugia, July 16th 1987. Degree Thesis title: Intersezioni di varietà algebriche dal punto di vista geometrico e topologico. Thesis advisers: Prof. U. Bartocci and Prof. L. Guerra.

Ph.D. Degree in Mathematics at University “La Sapienza” of Rome, November 5th, 1996. Ph.D. Thesis title: Classi di Weyl di varietà complesse. Thesis supervisor: Prof. A. Silva.

EDUCATION GRANTS

CNR: grant for graduants: March –July 1987. INDAM: graduated grants 1987/88 and 1988/89.

POSITIONS 1992-2003: Ricercatore (Assistant Professor), Faculty of Engineering, University of Perugia (Italy).

Current Position2004-today: Associate Professor (confirmed from 1.01.2007), Faculty of Engineering, University of Perugia (Italy).

VISITING POSITIONS Michigan Technological University, Houghton, USA, August 24th - September 3rd. MEMBER EDITORIAL BOARDPanAmerican Mathematical Journal: from July 2007 - today

RESEARCH INTERESTSCoding Theory, Finite Geometry, Algebraic Curves.

REFEREE FOR

Advances in Mathematics of CommunicationsDesigns, Codes and Cryptography Discrete MathematicsFinite Fields and Applications

Innovations in Incidence GeometryInternational Electronic Journal of GeometryJournal of Combinatorial DesignJournal of Combinatorial Theory Ser. AThe Journal of Combinatorial Mathematics and Combinatorial Computing

PARTECIPATION TO COURSES

26/7/1987 - 29/8/1987 Courses of Geometria Algebrica e Topologia Differenziale organized by Scuola Matematica Interuniversitaria (S.M.I.), Perugia, Italy

30/7/1989 – 19/8/1989 and 29/7/1991 - 16/8/1991 Courses of Geometria Differenziale organized by S.M.I. and by Scuola Normale Superiore, Cortona (Arezzo, Italy).

SUMMER SCHOOLS OF FINITE GEOMETRY:

Bella - Potenza, July, 27-31 1993;Bella - Potenza, September, 6-10 1993;Bienno - Brescia July, 17-23 1994;S. Felice del Benaco - Brescia, July 14-20 1996;Bella - Potenza, September, 2-6 1997.

MEMBER OF ORGANIZING COMMITTEE OG CONGRESSES

1. Combinatorics ’96, Assisi, September 8-14, 1996.2. GNSAGA, Perugia, November 6-8, 1997.3. Workshop Aspetti Computazionali in Matematica Discreta, nell’ambito del progetto “Applicazioni della Matematica per la Tecnologia e la Società”, sottoprogetto “Calcolo Simbolico” Perugia, May 3, 1997.4. IV Convegno del Dipartimento di Matematica e Informatica (IV) dedicato a Sauro Tulipani, Perugia, January 13-14, 2006.5. Joint International Meeting UMI-DMV, Perugia, June 18-22, 2007.6. CP 2011, XVII International Conference on Principles and Practice of Constraint Programming, Perugia, September 12-16, 2011.7. Combinatorics 2012, Perugia, September 9-15 2012.

Member UMI from 1999.

Member GNSAGA from 1995.

From 1996 she has coordinated the Projects Visiting Professors of Unita' Locale P.R.I.N. "Strutture Geometriche, Combinatoria e loro Applicazioni" (from 2010 with the designation "Geometrie di Galois e Strutture di Incidenza").Invited Visiting Professors: J. Barat, J.Bierbrauer, L.R.A. Casse, B. Cherowitzo, A.A. Davydov,

O. Heden, H. Kaneta, G. Kiss, P. Lisonek, T. Penttila, I. Pinneri, L. Storme, S. Zhou.

In particular from 1999 to 2012 she has been responsible of 9 projects for Visiting Professorsfinanced by GNSAGA.

SUPERVISOR of the research activity of PhD student Daniele Bartoli in the framework ofDottorato in Matematica e Informatica per l'elaborazione ela rappresentazione dell'informazione edella conoscenza (XXIV cycle). Thesis: "Construction and Classification of GeometricalStructures".

PARTECIPATION TO RESEARCH PROJECTS

- Progetto Nazionale di Ricerca Scientifica, Art.65 D.P.R. 382/1980 (ex quota 40%), “Geometria e Fisica” (coordinatore nazionale M. Francaviglia), 1989.

- Progetto di Ricerca di Ateneo Università degli Studi di Perugia, Art.65 D.P.R. 382/1980 (60%), “Geometria e Fisica” (coordinatore U. Bartocci), 1989, 1992/1997.

- Progetto Nazionale di Ricerca Scientifica, Art.65 D.P.R. 382/1980 (ex quota 40%), “Metodi Geometrici e Probabilistici in Sistemi Dinamici, Meccanica Statistica, Relatività e Teoria dei Campi” (coordinatore nazionale M. Francaviglia), 1991/1992, 1994.

- Progetto di Ricerca di Ateneo Università degli Studi di Perugia, Art.65 D.P.R. 382/1980 (60%), “Fondamenti di Geometria e Combinatorica” (coordinatore G. Faina), 1994/1997.

- Progetto Nazionale di Ricerca Scientifica, Art.65 D.P.R. 382/1980 (ex quota 40%), “Strutture Geometriche, Combinatoria e loro Applicazioni” (coordinatore nazionale G. Tallini), 1995.

- Progetto C.N.R. “Applicazioni della Matematica per la Tecnologia e la Società”, sottoprogetto “Calcolo Simbolico” (coordinatore nazionale A. Bonisoli), 1997.

- Progetto Cofinanziato Interuniversitario, D.M. del 23/4/1997, P.R.I.N. “Strutture Geometriche, Combinatoria e loro Applicazioni” (coordinatore nazionale F. Mazzocca), febbraio 1998 / febbraio 2000.

- Progetto di Ateneo Cofinanziato, Università degli Studi di Perugia “Modelli Matematici e Informatici per i Sistemi Intelligenti”, 1998/1999.

- Progetto “Intergovernmental Italian-Hungarian I-66 / 99 (promosso da G. Korchmaros e T. Szonyi).

- Progetto Cofinanziato Interuniversitario, D.M. del 18/10/1999, P.R.I.N. “Strutture Geometriche, Combinatoria e loro Applicazioni” (coordinatore nazionale G. Lunardon), novembre 1999 / novembre 2001.

- Progetto Cofinanziato Interuniversitario, D.M. del 12/11/2001, P.R.I.N. “Strutture Geometriche, Combinatoria e loro Applicazioni” (coordinatore nazionale G. Lunardon), novembre 2001 /novembre 2003.

- Progetto Cofinanziato Interuniversitario, D.M. del 23/10/2003, P.R.I.N. “Strutture Geometriche, Combinatoria e loro Applicazioni” (coordinatore nazionale G. Lunardon), novembre 2003 /novembre 2005.

- Progetto Nazionale MIUR “Lauree Scientifiche - Matematica - Umbria”, 2005/2006 - 2006/2007.

- Progetto Cofinanziato Interuniversitario, D.M. del 22/12/2005, P.R.I.N. “Strutture Geometriche, Combinatoria e loro Applicazioni” (coordinatore nazionale G. Lunardon), gennaio 2006 /gennaio 2008.

- Progetto Cofinanziato Interuniversitario, D.M. del 20/01/2010, P.R.I.N. “Geometrie di Galois e Strutture di Incidenza” (coordinatore nazionale G. Lunardon), gennaio 2010 /gennaio 2012.

RESEARCH ACTIVITY - DESCRIPTION

My research activity concerned Algebraic Curves, Galois Geometries, Coding Theory and in the earlier stages of my career also certain aspects of Mathematical Physics and Invariant Theory in Complex Differential Geometry.

The references between [] are listed below for typology.

1. ALGEBRAIC CURVES

A plane non-singular algebraic curve of degree n is called maximally symmetric if its group of automorphisms meets the corresponding bound over the complex numbers with equality. The objective of a series of papers was an attempt to classify the maximally symmetric plane non-singular algebraic curves of arbitrary degree defined over an algebraically closed field of characteristic zero.As a starting point particular values of n were studied: in [13] the cases n<20, n prime, were considered. In all those cases the most symmetric curves are projectively equivalent to the Fermat curve of equation x^n + y^n + z^n =0 whose group of automorphisms has order 6n^2.As a next step I considered the first non-prime degree n=6 which had not been completely classified (for n=4 one has the well-known Klein quartic). It is proved in [29] that the Wiman sextic is the unique maximally symmetric curve in this case. It is invariant under the alternating group A_6. This generalizes a result by H. Doi, K. Idei and H. Kaneta which is valid only over the complex numbers. Moreover in [29] it has been defined an immersion of PSL(2,7) and an immersion of A_6 in PGL(3,k), which allow to determine family of arcs with big automorphism group, projectively equivalent to the flexes of the Klein quartic or the Wiman's sextic, as described in the following paragraph. It should be noted that the first immersion has been used by G. Korchmaros ad L. Indaco (2011) and the second in [55] to determine infinite family of arcs with large automorphism group. The next step has been to generalize the result of [13] to every prime degree. In [AC20] and [61] I obtained a more general result: for every n different from 4 and 6, up to projectivity the Fermat curve is the unique non-singular algebraic plane curve which is maximally symmetric. This generalization requested a preliminary study of group theory in order to determine general properties for subgroups of PGL(3,k) such that the studied curves are invariant with respect to them.

Other types of works, concerning the connections between Finite Geometry and Algebraic curve theory are illustrated in the next section.

2. GALOIS GEOMETRIES

The first area of study in this field was the spectrum of cardinalities and the geometrical properties of complete arcs and caps in Galois geometries PG(r,q). An n-arc is a set of n points such that no more than (r+1) are on a hyperplane. It is complete if it is not contained in an (n+1)-arc. Their coding theoretic equivalent are MDS-codes. A cap in PG(r,q) is a set of points no three of which are collinear. It is complete if every point in space is collinear with some two points of the cap. Complete caps are of particular interest in coding theory as they are equivalent to q-ary linear quasi-perfect codes of redundancy (r+1) and minimum distance 4. It was in fact the link between information theory and Galois geometries which stimulated the research, initiated by B. Segre in 1950s, into the possible cardinalities of caps. In various cases this study has relied on intricate computer calculations. It is complicated to arrive at a satisfactory theoretical solution as the geometric, combinatorial and also group theoretic properties of finite planes vary considerably with the arithmetic properties of the order. This is a costumary phenomenon in number theory.Not many general results are known. In the case of the maximum possible cardinality this is a special case of the packing problem as formulated in the Congress “R.C. Bose Memorial Conference on Statistical Design and related Combinatorics” – Fort-Collins, Colorado. The same is true of the problem of determining the smallest cardinality of complete arcs or caps. A complete theoretical determination of the maximum cardinality of complete caps is known only in dimensions two and three: if the dimension is two the maximum size is q+1 (a conic) if q is odd and q+2 (hyperovals) if q is even; if the dimension is three the maximum size is q^2 + 1 (elliptic quadric).My research in this area has produced results in the following directions:

2.1. ARCS IN PROJECTIVE GALOIS PLANES

Each arc in PG(r,q) produces via a sequence of projections a plane arc. This leads to the situation that plane arcs are important in the general context of Galois geometries. My attack on the problems defined by B. Segre were based on algebraic geometry over finite fields and on combinatorial methods. The study of the connections between finite geometries and the theory of algebraic curves initiated in the 1950s with the fundamental works by Segre himself. This theory received an important impulse in the late 1980s by the Stöhr-Voloch theory concerning the number of rational points of an algebraic curve over a finite field. The central idea behind this theory is that curves with many rational points must have certain non-classical characteristics. The study of non-classical curves is the main instrument used to obtain the recent results on the possible cardinalities of complete arcs in the plane. More specifically the second largest cardinality of complete arcs is not known in general. Certain limits on this number are due to B. Segre, J.F. Voloch, J.A. Thas, J.W.P. Hirschfeld and G. Korchmáros. The proofs of these results are all based on the original idea of Segre's to associate to an n-arc in PG(2,q) an algebraic curve defined over GF(q) whose number of rational points is at least n(q-n+2). This algebraic curve is called the envelope of the arc.In [18] we study the components of the envelopes of sufficiently large arcs, in odd characteristic. In particular we concentrate on components that contain special points - rational points of the envelope which are neither singular nor flexes. The Stöhr -Voloch theory yields an estimate for the number of rational points which in turn allows to derive a limit for the cardinality of the complete arc. This limit depends only on q and on the last Frobenius order of the linear series intersected on this component by the conics of the plane. We also determine explicitly the classes of curves which can

occur as irreducible components of large complete arcs. Those curves are classical with respect to the linear series determined by the lines but neither classical nor Frobenius classical with respect to the conics. Those results have motivated further studies into those classes of curves, among others in the works of A.Aguglia-G.Korchmáros and of M.Giulietti where new results concerning their number of rational points are obtained.In [29] we construct highly symmetric arcs using the most symmetric curves of degrees 4 (the Klein quartic) and 6 (the Wiman sextic). More precisely it is shown that the sets of their flexes, of cardinality 24 and 72 respectively, in almost all characteristics form arcs admitting PSL(2,7) and A_6 as symmetry groups, respectively. Moreover those arcs turn out to be complete in all cases when complete arcs of that size are theoretically possible.In [55], using the immersion of A_6 in PGL(3,k), arcs of projective planes which are invariant under the action of A_6 are studied, obtaining for some values of q complete plane arcs of size less then the size of known complete arcs.

We focussed attention on the determination of complete arcs of small cardinaltity, the aim being to solve the problem of the smallest cardinality. In complete generality this problem appears to be untractable. This problem has received much attention: in the beginning of my research the knowledge of the minimal order in PG(2, q) was limited to cases q<=16 and general constructions were limited to cardinalities in the central part of the spectrum.In [6], [19], [37], [AC16] the minimum order was determined in PG(2, q) for 17<=q<=32. In cases q<= 27 we also determined the spectrum of complete arcs and completed the classification up to automorphisms.

In order to obtain those results it has been necessary to use a sophisticated system of electronic calculations based on the geometric-group theoretic properties. In fact the difficulties increase exponentially with q: to this aim we developed algorithms which use the geometric symmetry properties of the ambient space in order to reduce the number of cases to be considered and in order to avoid the construction of isomorphic copies of the same solution. In particular a hybrid algorithm has been developed which combines a classification step with an exhaustive search step based on backtracking. The advantage of this approach is in the fact that the informations concerning symmetries of the objects obtained in the classification process are used in the subsequent step. This allows the exclusion of cases which are equivalent to cases already considered.In [39] we develop a construction of arcs obtained by adding to a fixed highly saturating arc orbits of its stabilizer. This approach produces in particular in case q=37 complete arcs of cardinality smaller than previously known, case q=37 being the first open case in this respect. In [9] we present a geometric-group theoretic construction of an infinite family of small complete arcs which in certain cases yields the minimum cardinality.In [30] a large number of small arcs in PG(2,q), q<1000, are obtained, using randomized and heuristic algorithms and by manipulating the orbits of symmetry groups acting on PG(2, q). The results suggest certain general conjectures concerning the behaviour of the cardinalities close to the minimal cardinality.The refinement of a randomized greedy algorithm developed in [30] for the search of point sets with interesting extremal properties has led to search process for larger values of q (up to about 80,000). This led to the examples described in [AC19], [42], [52] [AC19], [AC22] and [56]: in many cases those examples produce new bounds on the minimum length of plane complete arcs. The bulk of the data produced by the heuristic algorithms has been analyzed in [AC24] where the asymptotic behaviour of the bound on the minimal length of complete plane arcs has been described.In [41] we develop a procedure for the construction of arcs of cardinality 6( ?q -1) in planes of order 2^{4h+2}. Those arcs turn out to be complete for q <=2^18.General constructions for complete arcs sharing many points with a conic have been the objective of [AC17], [52] e [60]. In [AC17] and [60] we find complete arcs in PG(2, 17) and in PG(2, 59) which

consist of (q + 3)/2 points on a conic and 4 and 3 points outside, respectively. Those are counterexamples to a result by G. Pellegrino from 1977 which states that for q=1 (mod 4) a complete arc with (q + 3)/2 points on a conic can have at most 2 points outside that conic.The ovals - plane (q + 1)-arcs - have the property to admit precisely one tangent in each point. In [AC11] and [46] we consider point sets in the plane with this property, the so-called semiovals. Those structures are of intrinsic geometric interest and also possess cryptographic applications (Batten 2000). We obtain characterization results, constructions and non-existence results. We also construct new examples of semiovals of large cardinality.

2.2. PLANE (n, k)-ARCS

One natural generalization of the concept of an arc in the plane is the notion of an (n, k)-arc (or arc of species k): a set of n points no k + 1 of which are collinear. A natural class of examples of an (n, k)-arc in PG(2, q) is constituted by the GF(q)-rational points of a plane algebraic curve defined over GF(q) without linear components. Here n is the number of such points and k is the degree of the curve. In general such an (n, k)-arc is incomplete. In [17] the interesting problem is addressed when those (n, k)-arcs will in fact be complete. This problem has initially been posed by Hirschfeld and Voloch. They gave a partial answer in case k = 3: apart from conics and cubics only a small number of examples were known, the Hermitian curves among them. The Hermitian curve is non-singular and not Frobenius classic with respect to the linear system determined by the lines of the plane. It was then natural to ask if the rational points of every non Frobenius classical curve form a complete (n, k)-arc. In [17] we give an affirmative answer under the additional hypothesis that the degree of the dual curve is less than q +1: this condition turns out to be satisfied by a certain sporadic curve defined over GF(8): it follows from a recent result due to J.Top that the curve mentioned above and the Fermat quartic over GF(9) are the only curves of genus 3 over GF(q) with more than 2q +6 rational points. In [17] we also present various examples of complete (n, k)-arcs derived from algebraic curves: we emphasize that in many of those cases geometrical structures with those parameters had not been known before.To each plane (n, k)-arc is canonically associated a linear code of length n and dimension 3: its error-correcting capability increases with the difference n-k (the redundancy). For fixed k one is therefore interested in the largest value of n for an (n, k)-arc (packing problem). Species 3 is particularly interesting as the (n, 3)-arcs in the plane correspond to linear NMDS-codes of dimension 3 (while an MDS code [n, k, d] has minimum distance meeting the bound n-k+1 an NMDS code has minimum distance n-k and its dual has minimum distance k) In [10], [26] the packing problem for arcs of species 3 in PG(2, 11) and PG(2, 13) is resolved. The largest maximal length of an (n, 3)-arc is 21 and 23, respectively. Also the classification of the examples of maximal order is given. This result is proved by an exhaustive search method which utilizes geometric relationships between arcs of species 2 and 3 ([AC7]) and which uses projective equivalence relationships of certain configurations. In [AC7] we obtain a lower bound on the maximal cardinality of an (m, r-1)-arc in the plane contained in an (n, r)-arc and determine the arcs of species 3 of large cardinality for q>=16. Some of those have maximal length under the additional hypothesis to contain arcs of fixed cardinality. Recently the packing problem for (n, k)-arcs has been resolved in PG(2, 16) as well ([AC23], [59]). As the difficulty of the problem is exponential in q we reached the solution by utilizing further geometric constraints concerning the structure of the solutions. This allows to prune the search tree in places which can be proved not to lead to a solution.Those definitive results concerning the packing problem yield new bounds on the existence of NMDS codes in PG(2, q), q = 11,13,16 and have allowed us via extension methods to prove the non-existence of other related NMDS codes ([15],[59]). In particular new examples of NMDS-codes of maximal length have been obtained in [15]. Further new results concerning existence and classification of NMDS-codes have been obtained in [33] and [AC1] starting from classification

results of (n, 3)-arcs. As special cases the classifications are obtained of all arcs of species 3 in PG(2, q); q = 7,8,9 in [14]. If we increase the dimension and make use of extension methods alone for the classification of NMDS-codes, the calculation times become unmanageable: as a first step to reduce the time we used the method to alternate between extension and classification methods. In order to reduce the computational complexity of the classification problem we introduced a preclassification step based on a readily computable invariant of the code [AC2]: this method allows us to obtain a partition of the search space in parts each of which contains just one or a small number of equivalence classes. The subsequent step of the classification turns out to have a linear cost in the applications. Moreover it has turned out to be advantageous for classification purposes to use a characterization of AMDS codes (those are more general than NMDS-codes in the sense that there are no restrictions on the dual code) as given in [AC4]. This has enabled us to determine the maximal of an NMDS-code in GF(q), q=7,8,9 for each dimension ([33]). Those results have resolved numerous cases which had been open in the specialized databases http://www.win.tue.nl/~aeb/voorlincod.html (not more updated), http://mint.sbg.ac.at and http://www-ma4.upc.es/~simeon/codebounds.html.A code is called optimal with respect to the Griesmer bound if n>(k-2)q+k. In [41] we give a detailed analysis of the (n, k)-arcs consisting of point orbits under the action of a power of the Singer cycle in PG(2, q): in certain cases the Griesmer bound is reached.

2.3. CAPS IN PG(r,q), r>2

As indicated earlier, Segre had emphasized the importance of the determination of the spectrum of the cardinalities of complete caps in PG(r,q) and in particular of the extreme values, from the point of view of the applications. For example the complete caps of minimal sizes give rise to quasi-perfect codes which for fixed redundancy have density as close as possible to the perfect codes.Before the mid-1990s only two infinite families of complete caps had been constructed, both in projective 4-spaces: one by B. Segre (1959) and the second by G. Tallini (1964). In 1996 in [4] we described an inductive construction of complete caps in projective spaces of arbitrary dimension over fields of characteristic 2. These caps have size q^{ r/2} in even dimension and 3q^{( r-1)/2} in odd dimension. In particular in dimension 4 those cardinalities are smaller than the cardinalities that had been known before (of order 2q^2).What definitive results on the minimal cardinality of complete caps are concerned, my research concentrated on dimensions 3 and 4; in 1998 the problem was resolved for q=5 and q=3 which had been the first open cases. In order to solve the problem in PG(3,7) and in PG(4,4) detailed theoretical considerations were necessary in order to reduce the calculation times for an exhaustive search. In particolar the result in PG(3,7), including the classification of the minimal examples has been possible only by using group-theoretical considerations as well as graph-theoretic Ramsey theory concerning the relation between the existence question of certain linear codes and of the arcs in question. In fact the classification of the arcs has made possible to exclude those linear codes. In [49] the non-existence of certain linear codes has been particularly useful for the introduction of certain constraints on the hyperplane sections of complete caps in PG(4,4).In [3], [11] e [12] geometric mechanisms are used for the construction of small complete arcs in PG(3,q), q odd, which in certain cases yield arcs of minimal cardinality.In [5], [7], [8], [24], [AC12], [42] e [49] we address the problem of the spectra of complete caps in various dimensions.

In [47] we improve upon the inductive method of [4] by broadening the class of plane caps which can be used as a basis of the inductive construction: the main result is the determination for q>8 and for each dimension r>2 of a new upper bound for the minimal cardinality of complete caps. In the

same paper the notion of a sum-point of a complete plane curve is introduced. This notion was helpful in the establishment of the new upper bound for q<= 2^15.

What the maximum cardinality of complete caps is concerned, new upper bounds are obtained [42]. In dimension 3 the maximum cardinality is known, so the emphasis is on the second largest cardinality: in [44] e [AC8] we construct a new infinite family of complete caps of size (q^{2}+ q+8)/2, q odd prime, q=2(mod 3), q>=11. This yields a new lower bound on the second cardinality. A slight variant of this construction method yields in case q=5 one of the two complete 20-caps of the second largest cardinality.

2.4. SATURATING SETS

The r-saturating sets of a projective space generalize the concept of a complete cap. Technically speaking a subset S of PG(r, q) is called s-saturating if s is the minimum integer such that for every point P of PG(r, q) there exists a subspace generated by some s + 1 points of S which contains P. In case s=1 this is equivalent to the fact that the bisecants of S cover the ambient space. Of particular interest is the construction of minimal saturating sets (not containing proper subsets which are saturating) of small cardinality, in view of the links to covering codes and to other area of Combinatorics like the design of experiments and the degree/diameter- problem in graph theory.The problem of determining the minimal cardinality of saturating sets turns out to be more complex than the case of complete caps due to the fact that there are less geometric constraints.The best known general upper bound on the minimal cardinality concerns 1-saturating sets in the plane and is derived from statistical estimations: this bound is however quite far from a lower bound which is obtained by a simple counting argument. What exact values of the minimal cardinality in the plane is concerned, in [20] and [AC21] the problem has been resolved for q<=23, including the classification of the examples. The complete spectrum of examples has been determined for q<=13: in [22] estimations are obtained for certain extremal parameters pertaining to minimal saturating sets in PG(n, q). A concept of saturating density is introduced which allows to obtain new lower bounds on the smallest minimal saturating sets. In the same paper we also determine the largest cardinality of a minimal 1-saturating set. Constructions of minimal 1-saturating sets in PG(r, 2) are described in [AC3] and [36]. Some of these constructions yield sets of a particularly symmetric structure in terms of internal lines, polygons and orbits of the symmetry groups. As an example we studied an 11-set in PG(4, 2) named central pentagon. Starting from examples of those constructions infinite families of 1-saturating sets in spaces of growing dimension are obtained. In small dimensions the complete classification of the 1-saturating minimal sets is given.To an s-saturating n-set in PG(r, q) is canonically associated an [n,n-(r+1)]_R covering code of covering radius R = s+1: In [23] new constructions are given for infinite families of covering codes with R = 2; 3 and growing dimension. The basis for those inductive constructions consists of the saturating sets constructed in the previous articles. The interest of those constructions stems from the fact that codes with low covering density are obtained. In [AC6] and [31] the concept of a locally optimal covering code is introduced. It corresponds to the minimality property of saturating sets. Constructions of infinite families of such codes are given. Classification results are derived for those codes in ”small geometries” (for small values of the codimension and of q), using algorithms written in Magma which perform a search in width, utilizing projective equivalence and preclassification steps based on invariants like for example the minimum distance. New upper and lower bounds on the minimal and maximal lengths are obtained. In [AC13], [AC14] and [50] we continue the study of r-saturating sets in spaces of dimension exceeding 3. A new instrument has been the utilization of the notion of a strong blocking set which is a specialization of the classical notion of a blocking set of species n. In particular we obtain, by use of a concatenation construction, new infinite families of saturating sets of small cardinality in arbitrary dimension which yield q-ary

covering codes of asymptotic covering density higher than had been known before.In [AC25] a concept of a multiple s-saturating set is introduced. It has been proved that in the linear case those are the geometric counterpart of the multiply covering codes with MCF. One application of those codes is in the football pool problem where the aim is to reduce the previsions needed to multiply a victory. For such MCF codes we introduce a concept of multiple saturating density as a quality characteristic: in order to save resources one aims at those with lowest possible density. In geometric language this comes down to multiple saturating sets of minimum length.In the same work constructions are obtained of multiple 1-saturating sets of small cardinality which improve on an upper bound which can be derived via probabilistic results on classical 1-saturating sets. This paper belongs in the context of current research where recently general construction procedures of saturating multiple sets have been obtained using concatenation and translation caps.

2.5. CYCLIC MODEL of PG(r,q) and ARCS

Studying the additive inverse of lines in the cyclic model of PG(r,q), in [16] we generalized a result of M.Hall which states that in the cyclic model of the projective Galois plane the additive inverse of a line is a conic; in particular we showed that for the additive inverse of a line there are two possibilities: either it is a (q + 1)-arc or it is contained in a hyperplane. In dimension 3 the result is stronger as for the second possibility we obtain a line. This result has then been generalized to spaces of dimension larger than 3.

2.6. SPREADS E BLOCKING SETS

The problem of the determination of maximal partial spreads in PG(3, q) (families of mutually skew lines such that any line in the space PG(3, q) intersects at least one line of the family non-trivially) has been considered. Originally this problem has been studied by Dale Mesner in 1967. The currently best known upper bound is due to Aart Blokhuis as a consequence of his results on blocking sets. In his study of the case PG(3, 7) O. Heden proved that this upper bound cannot be improved upon in general. The first open case for odd q was q = 9 with possible cardinalities 75 and 76. In [40] the second possibility has been ruled out: in a preliminary paper [38] the corresponding blocking sets in PG(2, 9) have been determined and classified. This has been used as point of departure for the geometric study of the possible holes (points in the ambient space not covered by a line of the spread). The result has been obtained invoking a theorem by Blokhuis-Metsch concerning the weights of affine subspaces. In [L1] with analogous approach, also the first possibility has been ruled out, solving definitively the problem.

In the line of research concerning blocking sets we studied partitions of PG(2, q); specifically we aimed at determining the maximal number of disjoint minimal blocking sets. In particular it has been shown in [35] that a recent conjecture of M. Kriesell according which planes of non-square order q contain q/2 disjoint blocking sets may not be best possible. In fact we found 5 disjoint minimal blocking sets for q = 8.

In parallel we studied blocking sets in Mobius planes. There are no general constructions and only a small number of known bounds: L. Lovasz has proved that each Mobius plane of order q contains a blocking set of order less than 3qlogq whereas Bruen and Rothschild proved that for q>=9 the cardinalities are greater than 2q-1. In [P27] we determined the spectrum of the cardinalities for small values of q and showed that for each natural number C there is a constant q(C) for each Mobius plane of order q>q(C) the cardinality of a blocking set has at least 2q + C points.

3. CODING THEORY

My research in this area, above the results presented in the previous sections related to Galois geometries, has been in various directions.

3.1 TOPOLOGY OF NETWORKS

In [21] the problem of packet switched networks has been attacked. We translated it into design theoretic language. The solution of this problem involves weighted (n,k)-arcs in projective planes and 3-dimensional codes. The main problem concerned maximal coverings. A construction method for maximal coverings has been presented which then turned out to be uniquely determined. This resolves a problem which according to a statement of one of the leading experts in this area (C.J. Colbourn , 1999) seemed inaccessible as it seemed to contain the solution of the packing problem of plane arcs of arbitrary species. The solution of the problem was considered to be particularly relevant and has been the object of international research efforts for almost two decades. In those attempts various methods were used in order to translate the problem into different areas: graph theory [J.C.Bermond et al. 1983], computer aided numerical analysis and design theory [B.Yener et al., 1997], theory of projective planes [C. J. Colbourn, 2002]. Only in [21] the level of generality is reached which allows to establish a complete solution in a generic case. It also leads to links with coding theory which had not been known before and to a new type of a design-theoretic problem.More novel aspects in the approach of [21] are the use of the graph-theoretical concept, the fractional matching number and the use of linear programming which allows to demonstrate optimality in the relevant generic situation. Finally in [21] the design-theoretic formulation leads to a new type of extremal structures which generalizes the notion of an almost projective plane: significative examples are found in the paper.

3.2 NMDS CODES

This class of codes (both the code and its dual of Singleton defect 1) are the most valuable after the MDS codes.In fact the problem of finding good linear codes can be described as maximizing the minimum distance when length and dimension are given. It is known that the minimum distance cannot exceed the Singleton bound and MDS codes are precisely those which achieve equality. Essentially those cannot exist for length larger than n+2. It follows that the next problem along this line consists in the construction of codes of Singleton defect 1.In [AC5] and [28] we find a family of highly symmetric linear codes whose automorphism group contains the direct product of a Galois group and a group of permutations acting regularly. The family contains many classical codes (the extended Hamming code, the hexacode, the Golaycodes, the Pless codes) and also some new codes defined over GF(4) and GF(8) with extremely good parameters. The family can be extended in a natural way to codes defined over a ring. In fact the Z_4-linear octacode which yields the famous Nordstrom-Robinson code belongs to the family as well.The paper contains also the algebraic-geometric description of one of the two maximal NMDS codes over GF(8) (which are classified in [15]). This is the code which possesses an unusually large group of symmetries (direct product of an A_4 and an elementary abelian group of order 16). The description makes use of the combinatorial notion of an ordered design. The results closely connected to (n,k)-arcs have been illustrated in the previous section.

3.3 CODICI LDPC

In [54] new classes of LDPC codes are constructed, in some cases with new parameters. This has been achieved via a geometric approach based on the concept of a configuration. In view of the efficiency of the decoding algorithm the Tanner graph associated to the code is not allowed to contain short cycles. This poses the mathematical problem to construct bipartite graphs with large girth. In combinatorial terms this comes down to the construction of symmetric configurations.Group-theoretic and combinatorial methods have been used in [AC10] and [AC15] for the construction of new classes of configurations.In particular in [AC15] we used the Golomb rulers and the modular Golomb rulers as key instruments.This led to many new parameters, in particular in the cyclic case; new upper bounds have been derived on the minimum integer E(k) (Ec(k)) such that for every v > E(k) (Ec(k)) there exists a simmetric v_k-configuration (cyclic simmetric).In the area of symmetric graphs we studied in [57] the unitary graphs. Their classification constitutes the main result of the paper. Those graphs are a family of symmetric graphs whose vertices are the flags of the Hermitian unital where the adjacency relation is determined by the underlying field. They admit the unitary group as a group of automorphisms. Those graphs play a significant role in the classification of the symmetric graphs with complete quotients such that the corresponding incidence relation is 2-transitive on the points.

3.4 ADDITIVE CODES

The notion of an additive code is a vast range generalization of the classical concept of a linear code. In the most general sense they are linear over a finite abelian group written additively. In my research this concept is used in a slightly restricted variant where the alphabet is considered to be a vector space and linearity holds only with respect to the underlying field.In the geometric interpretation the theory of linear codes is equivalent to the theory of point sets in projective Galois spaces and their distribution on hyperplanes. The geometric interpretation of additive q_2-ary q-linear codes (this is a parametric subfamily) is the theory of lines in space and their distribution on hyperplanes. This restricted concept is in itself a far-reaching generalization of the concept of a linear code. In fact it contains the theory of quantum stabilizer codes as a special case. In [AC9] we use the geometric description to prove the non-existence of an additive [12,7,5]_4-code. The complete determination of the parameters of quaternary additive codes of length 12 is in [45], again using geometric language. This paper also contains the construction of a new optimal code of length 13. In [48] we present another non-existence result which in geometric terms is equivalent to the statement that for every system of 12 lines in PG(8, 2) there is a hyperplane containing more than 5 of those lines.

3.5 QUANTUM CODES

In those last years I started studying the theory of quantum codes. The research on quantum computations, although still in an embryonal state, is extremely active, supported also by national governments and military agencies.The quantum error-correcting codes are very significant in protecting from information errors in a future quantum computer. The construction of a quantum computer currently is the aim of various research groups. Potentially a quantum computer may be able to solve problems much faster than any classical computer. One popular example is the factorization of composite numbers, a problem

which is at the heart of the most common cryptographical algorithms.The fundamental work of CRSS (1998) translates the problem of error correction in quantum computations into the language of coding theory over finite fields (quantum stabilizer codes, a special case of additive codes). In [43] we take another step forward by translating the problem of quantum error-correction into a geometric language: binary quantum codes and more in general quaternary additive codes are described in terms of systems of points and lines in binary Galois spaces. This approach has shed new light on some classical quantum code families, has allowed new constructions and has led to relations with objects like quadrics and APN functions (a basic concept in the cryptographical theory of S-boxes). This geometric point of view has allowed us in [43] to solve certain problems concerning the existence of optimal quantum codes, in particular the construction of quantum codes with new parameters. An analogous geometric approach was used in [51] to demonstrate the non-existence of a [[13, 5, 6]] quantum stabilizer code, thus solving a problem which has been open for years. The proof relies not only on systems of points and lines in the ambient space PG(7,2) but also in certain subspaces and factor spaces. As a consequence several open problems have been removed from the database M. Grassl, http://www.codetables.de. In [AC18], [L2], [53], we study linear quantum codes with parameters [[n, n -10, 4]] via their geometric counterparts, caps in PG(4, 4) satisfying the property that the intersection cardinality with any hyperplane always has the same parity as the cardinality n. In particular we determine the spectrum of those caps and we obtain a complete classification of the extremal examples. Those examples have been used to obtain inductive definitions of infinite families of quantum caps in higher dimension.General constructions for quantum codes over arbitrary finite fields are described in [58]. One such construction generalizes a theorem from the fundamental paper by CRSS. The central part of our paper consists of the study of quantum caps in quaternary projective spaces. Various recursive constructions are obtained.

4. MATHEMATICAL PHYSICS

In the early stages of the career I undertook a study of explicit formulas for the volume of a family of quadratic caps. This study was motivated by an attempt to estimate the volumes of certain tumorsbased on three dimensions which are known via direct measurements ([1]).In the sequel the activity went into the direction of experimental confirmations of special relativity.In particular the Michelson-Morley experiment has been analyzed ([2]) in order to point out certain aspects of the physical-mathematical analysis which had been neglected. This led to the discovery of various errors in the standard textbooks and to the description of a rigorous theoretical treatment of this experiment.

5. INVARIANTS IN COMPLEX DIFFERENTIAL GEOMETRY

The doctoral thesis is in this research area ([TD]). It is focussed on the study of a certain type of characteristic forms, the Weyl forms, which are associated to a projective connection on complex varieties. In particular certain families of examples have been obtained for which Weyl classes are significant. Their relation to the Chern classes have been analysed. The original results have beenexposed in [RT1].

PUBLICATIONS

1) PAPERS ON JOURNALS OR IN COURSE OF PUBLICATIONS

[61] F. Pambianco, Characterization of the Fermat curve as the most symmetric non-singular algebraic plane curve, submitted.

[60] F. Pambianco, D. Bartoli, A. A. Davydov, G. Faina, and S. Marcugini, A 3-cycle construction of complete arcs sharing (q+3)/2points with a conic, submitted.

[59] F. Pambianco, D. Bartoli, and S. Marcugini, The non-existence of some NMDS codes, Designs, Codes and Cryptography, accepted.

[58] F. Pambianco, D. Bartoli, J. Bierbrauer, Y. Edel, G. Faina, and S. Marcugini, The structure of quaternary quantum caps, Designs, Codes and Cryptography, DOI: 10.1007/s10623-013-9796-5

[57] F. Pambianco, M. Giulietti, S. Marcugini, and S. Zhou, Unitary graphs and classification of a family of symmetric graphs with complete quotients, Journal of Algebraic Combinatorics, DOI: 10.1007/s10801-012-0422-9

[56] F. Pambianco, D. Bartoli, A. A. Davydov, G. Faina, and S. Marcugini, New upper bounds on the smallest size of a complete arc in the plane PG(2,q), Journal of Geometry, DOI: 10.1007/s00022-013-0154-6

[55] F. Pambianco, M. Giulietti, G. Korchmáros, and S. Marcugini, Transitive A_6-invariant k-arcs in PG(2,q), Designs, Codes and Cryptography, ISSN: 0925-1022, DOI: 10.1007/s10623-012-9619

[54] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, Some Combinatorial Aspects of Constructing Bipartite-Graph Codes, Graphs and Combinatorics, 29 (2013) 187-212, ISSN: 0911-0119, DOI: 10.1007/s00373-011-1103-5

[53] F. Pambianco, D. Bartoli, G. Faina, and S. Marcugini, New quantum caps in PG(4,4), Journal of Combinatorial Design, 20 (2012), 448-466, ISSN: 1063-8539

[52] F. Pambianco, D. Bartoli, A. A. Davydov, G. Faina and S. Marcugini, On sizes of complete arcs in PG(2; q), Discrete Mathematics, 312 (2012) 680-698, ISSN: 0012-365X, DOI: 10.1016/j.disc.2011.07.002

[51] F. Pambianco, J. Bierbrauer, R.D. Fears, and S. Marcugini, The non existence of a [[13,5,4]]-quantum stabilizer code, IEEE Transactions on Information Theory, 57 (2011), 4788-4793, ISSN: 0018-9448

[50] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, Linear nonbinary covering codes and saturating sets in projective spaces, Advanc. Math. Commun. 5 (2011), 119-147, ISSN: 1930-5346, DOI: 10.3934/amc.2011.5.119

[49] F. Pambianco, D. Bartoli, A.A. Davydov, and S. Marcugini, The minimum order of complete caps in PG(4; 4), Advanc. Math. Commun. 5 (2011), 37-40, ISSN: 1930-5346, DOI:

10.3934/amc.2011.5.37

[48] F. Pambianco, J. Bierbrauer, and S. Marcugini, A geometric non-existence proof of an extremal additive code, Journal of Combinatorial Theory, Series A,vol. 117 (2010), 128-137, ISSN: 0097-3165

[47] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, New inductive constructions of complete caps in PG(N,q), q even, Journal of Combinatorial Design, 18 (2010) 177-201, ISSN: 1063-8539, DOI: 10.1002/jcd.20230

[46] F. Pambianco, Gy. Kiss, and S. Marcugini, On the spectrum of the sizes of semiovals in PG(2,q), q odd, Discrete Mathematics, 310 (2010), 3188-3193, ISSN: 0012-365X, DOI: 10.1016/j.disc.2009.07.024.

[45] F. Pambianco, J. Bierbrauer, Y. Edel, G. Faina and S. Marcugini, Short additive quaternary codes, IEEE Transactions on Information Theory, 55 (2009), 952-954, ISSN: 0018-9448

[44] F. Pambianco, A. A. Davydov, and S. Marcugini, Complete (q^2 + q + 8)/2-caps in the spaces PG(3,q) q=2 (mod 3) an odd prime, and a complete 20-cap in PG(3,5), Designs, Codes and Cryptography, 50 (2009), 359-372, ISSN: 0925-1022

[43] F. Pambianco, G. Faina, M. Giulietti, S. Marcugini, and J. Bierbrauer, The geometry of quantum codes, Innovations in Incidence Geometry, 6-7 (2009), 53-71, ISSN: 1781-6475

[42] F. Pambianco, A. A. Davydov, G. Faina, and S. Marcugini, On sizes of complete caps in projective spaces PG(n,q) and arcs in planes PG(2,q), Journal of Geometry, 94 (2009), 31-58, ISSN: 0047-2468

[41] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, On sharply transitive sets in PG(2,q), Innovations in Incidence Geometry, 6-7 (2009), 139-151, ISSN: 1781-6475

[40] F. Pambianco, O. Heden, S. Marcugini, and L. Storme, On the non-existence of a maximal partial spread of size 76 in PG(3,9) Ars Combinatoria, 89 (2008), 369-382, ISSN: 0381-7032

[39] F. Pambianco, S. Marcugini, and P. Lisonek, Constructions of Small Complete Arcs with Prescribed Symmetry, Contributions to Discrete Mathematics, 3 (2008), 14-19, ISSN: 1715-0868

[38] F. Pambianco and L. Storme, Minimal blocking sets in PG(2,9), Ars Combinatoria, 89 (2008), 223-234, ISSN: 0381-7032

[37] F. Pambianco, S. Marcugini, and A. Milani, Complete arcs in PG(2,25): the spectrum of the sizes and the classification of the smallest complete arcs, Discrete Mathematics, 307 (2007), 739-747, ISSN: 0012-365X

[36] F. Pambianco, A. A. Davydov, and S. Marcugini, Minimal 1-saturating sets and complete caps in binary projective spaces, Journal of Combinatorial Theory, Series A, 113 (2006), 647-663, ISSN: 0097-3165

[35] F. Pambianco, J. Barat, S. Marcugini, and T. Szonyi, Note on disjoint blocking sets in Galois planes, Journal of Combinatorial Design, 14 (2006), 149-158, ISSN: 1063-8539

[34] F. Pambianco, J. Bierbrauer, and S. Marcugini, The smallest size of a complete cap in PG(3,7), Discrete Mathematics, 306 (2006), 1257-1263, ISSN: 0012-365X

[33] F. Pambianco, S. Marcugini, and A. Milani, Classification of linear codes exploiting an invariant, Contributions to Discrete Mathematics, 1 (2006), 1-7, ISSN: 1715-0868

[32] F. Pambianco, A. A. Davydov, and G. Faina, Constructions of Small Complete Caps in Binary Projective Spaces, Designs, Codes and Cryptography, 37 (2005), 61-80, ISSN: 0925-1022

[31] F. Pambianco, A. A. Davydov, G. Faina, and S. Marcugini, Locally Optimal (Nonshortening) Linear Covering Codes and Minimal Saturating Sets in Projective Spaces, IEEE Transactions on Information Theory, 51 (2005), 4378-4387, ISSN: 0018-9448

[30] F. Pambianco, A. A. Davydov, G. Faina, and S. Marcugini, Computer search in projective planes for the sizes of complete arcs, Journal of Geometry, 82 (2005), 50-62.

[29] F. Pambianco, H. Kaneta, and S. Marcugini, On arcs and curves with many automorphisms, Mediterranean Journal of Mathematics, 2 (2005) 71-102, ISSN: 1660-5446

[28] F. Pambianco, S. Marcugini, and J. Bierbrauer, A family of highly symmetric NMDS codes, IEEE Transactions on Information Theory, 51 (2005) 3665-3668, ISSN: 0018-9448

[27] F. Pambianco, Gy. Kiss, and S. Marcugini, On blocking sets of inversive planes, Journal of Combinatorial Design, 13 (2005) , 268-275, ISSN: 1063-8539

[26] F. Pambianco, S. Marcugini, and A. Milani, Maximal (n,3)-arcs in PG(2,13), Discrete Mathematics 294 (2005), 139-145, ISSN: 0012-365X

[25] F. Pambianco, S. Marcugini, and A. Milani, Classification of the (n,3)-arcs in PG(2,7), Journal of Geometry, 80 (2004), 179-184.

[24] F. Pambianco, A. A. Davydov, and S. Marcugini, Complete caps in projective spaces PG(n,q), Journal of Geometry, 80 (2004), 23-30.

[23] F. Pambianco, A. A. Davydov, and S. Marcugini, Linear codes with covering radius 2,3 and saturating sets in projective geometry, IEEE Transactions on Information Theory, 50 (2004), 537-541, ISSN: 0018-9448.

[22] F. Pambianco, A. A. Davydov, and S. Marcugini, On saturating sets in projective spaces, Journal of Combinatorial Theory, Series A, 103 (2003) 1-15, ISSN: 0097-3165

[21] F. Pambianco, J. Bierbrauer, and S. Marcugini, Projective planes, coverings and a network problem, Designs, Codes and Cryptography, 29 (2003), 71-89, ISSN: 0925-1022.

[20] F. Pambianco and S. Marcugini, Minimal 1-saturating sets in PG(2,q), Australasian Journal of Combinatorics, 28 (2003) 161-169, ISSN: 1034-4942.

[19] F. Pambianco, S. Marcugini, and A. Milani, Minimal complete arcs in PG(2,q), q<=29, The Journal of Combinatorial Mathematics and Combinatorial Computing, 47 (2003), 19-29.

[18] F. Pambianco, M. Giulietti, F. Torres, and E. Ughi, On large complete arcs: odd case, Discrete Mathematics, 255 (2002) 145–159.

[17] F. Pambianco, M. Giulietti, F. Torres, and E.Ughi, On complete arcs arising from plane curves, Designs, Codes and Cryptography, 25 (2002), 237-246.

[16] F. Pambianco, G. Faina, Gy. Kiss, and S. Marcugini, The cyclic model for PG(n,q) and a costruction of arcs, European Journal of Combinatorics, 23 (2002), 31-35.

[15] F. Pambianco, S. Marcugini, and A. Milani, NMDS codes of maximal length over GF(q), 8<= q<= 11, IEEE Transactions on Information Theory 48 (2002), 963-966.

[14] F. Pambianco, S. Marcugini, and A. Milani, Classification of the [n,3,n-3]q NMDS codes over GF(7), GF(8) and GF(9), Ars Combinatoria 61 (2001), 263-269.

[13] F. Pambianco, H. Kaneta, and S. Marcugini, The most symmetric non-singular plane curves of degree n <= 20, I, Geom. Dedicata 85 (2001) 317-334.

[12] F. Pambianco and E. Ughi, A class of k-caps having k-2 points in common with an elliptic quadric and two points on an external line, Australasian J. of Comb. 21 (2000), 299-310.

[11] F. Pambianco, A class of complete k-caps of small cardinality in projective spaces over fields of characteristic three, Discrete Math. 208/209 (1999), 463-468.

[10] F. Pambianco, S. Marcugini, and A. Milani, Maximal (n,3)-arcs in PG(2,11), Discrete Math. 208/209 (1999), 421-426.

[9] F. Pambianco and G. Faina, On some 10-arcs for deriving the minimum order for complete arcs in small projective planes, Discrete Math., 208/209 (1999), 261-271.

[8] F. Pambianco and G. Faina, On the spectrum of the values k for which a complete k-cap in PG(n,q) exists, J. Geom. 62 (1998), 84-98.

[7] F. Pambianco, G. Faina, S. Marcugini, and A. Milani, F. P., The sizes of the complete k-caps in PG(n,q), for small q and 3 <= n <= 5, Ars Combinatoria 50 (1998), 235-243.

[6] F. Pambianco, G. Faina, S. Marcugini, and A. Milani, The spectrum of the values k for which there exists a complete k-arc in PG(2,q) for q<=23, Ars Combinatoria 47 (1997), 3-11.

[5] F. Pambianco and G. Faina, Small complete k-caps of PG(r,q), r >= 3, Discrete Math. 174 (1997), 117-123.

[4] F. Pambianco and L. Storme, Small complete caps in spaces of even characteristic, J. Comb. Theory, Ser. A, 75 No.1 (1996), 70-84.

[3] F. Pambianco and G. Faina, A class of complete k-caps in PG(3,q) for q an odd prime, J. Geom. 57 (1996), 93-105.

[2] F. Pambianco and M. Mamone Capria, On the Michelson-Morley experiment, Foundations of Physics 24 (1994), 885-899.

[1] F. Pambianco and U. Bartocci, Sulla stima del volume di certe calotte, Archimede 4 (1989), 186-195.

2. ARTICLES IN BOOKS

[L2] F. Pambianco, D. Bartoli, S. Marcugini, and J. Bierbrauer, Geometric Constructions of quantum codes. In Error-Correcting Codes, Cryptography and Finite Geometries, A.A. Bruen and D. Wehlau Ed. AMS, Series: Contemporary Mathematics, vol. 523 (2010), p. 149-154. ISBN/ISSN: 9780821849569

[L1] F. Pambianco, G. Faina, O. Heden, and S. Marcugini, The maximal size of a maximal partial spread in PG(3,9). In: Trends in Incidence and Galois Geometries: a Tribute to Giuseppe Tallini, Series Quaderni di Matematica, vol. 19 (2010), p. 77-112, Roma: Aracne, ISBN/ISSN: 9788854835719

3. PAPERS in PROCEEDINGS (with referee)

[AC25] F. Pambianco, D. Bartoli, A. A. Davydov, M. Giulietti, and S. Marcugini, Multiple coverings of the farthest-off points and multiple saturating sets in projective spaces, Proceedings of ACCT 2012, Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory, Pomorie, Bulgaria, June 2012, 53-59, ISSN 1313-423X.

[AC24] F. Pambianco, D. Bartoli, A. A. Davydov, and S. Marcugini, New type of estimations for the smallest size of complete arcs in PG(2,q), Proceedings of ACCT 2012, Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory, Pomorie, Bulgaria, June 2012, 67-72, ISSN 1313-423X.

[AC23] F. Pambianco, D. Bartoli, and S. Marcugini, The maximum and minimum sizes of complete (n,3)-arcs in PG(2,16), Proceedings of ACCT 2012, Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory, Pomorie, Bulgaria, June 2012, 77-82, ISSN 1313-423X.

[AC22] F. Pambianco, D. Bartoli, A. A. Davydov, G. Faina, and S. Marcugini, New upper bounds on the smallest size of a complete arc in the plane PG(2,q), Proceedings of ACCT 2012, Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory, Pomorie, Bulgaria, June 2012, 60-66, ISSN 1313-423X.

[AC21] F. Pambianco, D. Bartoli, G. Faina, and S. Marcugini, Classification of minimal 1-saturating sets in PG(2,q), q <=23, Proceedings of ACCT 2012, Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory, Pomorie, Bulgaria, June 2012, 73-76, ISSN 1313-423X.

[AC20] F. Pambianco, The Fermat curve x^n + y^n + z^n: the most symmetric non-singular algebraic plane curve, Proceedings of ACCT 2010, Twelfth International Workshop of Algebraic and Combinatorial Coding Theory, Novosibirsk, Russia, September 2010, 245-250, ISBN: 9785861341745

[AC19] F. Pambianco, A. A. Davydov, G. Faina, and S. Marcugini, New sizes of complete arcs in PG(2; q), Proceedings of ACCT 2010, Twelfth International Workshop of Algebraic and

Combinatorial Coding Theory, Novosibirsk, Russia, September 2010, 103-108, ISBN: 9785861341745.

[AC18] F. Pambianco, D. Bartoli, and S. Marcugini, The spectrum of linear pure quantum [[n,n-10,4]]-codes, Proceedings of ACCT 2010, Twelfth International Workshop of Algebraic and Combinatorial Coding Theory, Novosibirsk, Russia, September 2010, 5-11, 31-36, ISBN: 9785861341745.

[AC17] F. Pambianco, A. A. Davydov, and S. Marcugini, A geometric construction of complete arcs sharing (q + 3)/2 points with a conic, Proceedings of ACCT 2010, Twelfth International Workshop of Algebraic and Combinatorial Coding Theory, Novosibirsk, Russia, September 2010, 109-115, ISBN: 9785861341745.

[AC16] F. Pambianco, S. Marcugini, and A. Milani, Minimal complete arcs in PG(2,q), q<= 32, Proceedings of ACCT 2010, Twelfth International Workshop of Algebraic and Combinatorial Coding Theory, Novosibirsk, Russia, September 2010, 217-222, ISBN: 9785861341745.

[AC15] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, On the spectrum of possible parameters of symmetric configurations, Proceedings of XII International Symposium on Problems of Redundancy in Information and Control Systems, St. Petersburg, Russia, May 26-30, 2009, 59-64, ISBN: 9785808804470

[AC14] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, Linear covering codes of radius 2 and 3, Proceedings of the Workshop “Coding Theory Days in St. Petersburg”. Saint-Petersburg, Russia, October 2008, 12-17, ISBN: 9785808803787

[AC13] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, Linear covering codes over nonbinary finite fields, Proceedings of ACCT 2008, Eleventh International Workshop of Algebraic and Combinatorial Coding Theory, Pamporovo, Bulgaria, June 16-22 2008, 70-75, ISSN: 1313-423X

[AC12] F. Pambianco, A. A. Davydov G. Faina, and S. Marcugini, On the spectrum of sizes of complete caps in projective spaces PG(n,q) of small dimension, Proceedings of ACCT 2008, Eleventh International Workshop of Algebraic and Combinatorial Coding Theory, Pamporovo, Bulgaria, June 16-22 2008, 57-62, ISSN: 1313-423X

[AC11] F. Pambianco Gy. Kiss, and S. Marcugini, Semiovals in projective planes of small order, Proceedings of ACCT 2008, Eleventh International Workshop of Algebraic and Combinatorial Coding Theory. Pamporovo, Bulgaria, June 16-22 2008, 151-154, ISSN: 1313-423X

[AC10] F. Pambianco, A. A. Davydov, M. Giulietti, and S. Marcugini, Symmetric configurations for bipartite-graph codes, Proceedings of ACCT 2008, Eleventh International Workshop of Algebraic and Combinatorial Coding Theory, Pamporovo, Bulgaria, June 16-22 2008, 63-69, ISSN: 1313-423X

[AC9] F. Pambianco, J. Bierbrauer, G. Faina, and S. Marcugini, Additive quaternary codes of small length, Proceedings of ACCT 2006, Tenth International Workshop on Algebraic and Combinatorial Coding Theory, Zvenigorod, Russia, September 3-9 2006, 15-18.

[AC8] F. Pambianco, A. A. Davydov, and S. Marcugini, Complete(q̂2 + q + 8)/2-caps in the projective spaces PG(3,q), with odd prime q=2 (mod 3), Proceedings of ACCT 2006, Tenth

International Workshop on Algebraic and Combinatorial Coding Theory, Zvenigorod, Russia, September 3-9 2006, 76-80.

[AC7] F. Pambianco, J. Bierbrauer, G. Faina, and S. Marcugini, On the structure of the (n,r)-arcs in PG(2,q), Proceedings of ACCT 2006, Tenth International Workshop on Algebraic and Combinatorial Coding Theory. Zvenigorod, Russia, September 3-9 2006, 19-23.

[AC6] F. Pambianco, A. A. Davydov G. Faina, and S. Marcugini, Locally optimal covering codes and minimal saturating sets, Proceedings of OC 2005, Fourth International Workshop on Optimal Codes and Related Topics, Pamporovo, Bulgaria, June 17-23 2005, 114-120, ISBN: 9548986183

[AC5] F. Pambianco, J. Bierbrauer, and S. Marcugini, Some highly symmetric codes, Proceedings of the Fourth International Workshop on Optimal Codes and Related Topics. Pamporovo, Bulgaria, June 17-23 2005, 14-19, ISBN: 9548986183

[AC4] F. Pambianco and S. Marcugini, AMDS codes of small dimension, Proceedings of ACCT 2004, Ninth International Workshop on Algebraic and Combinatorial Coding Theory, Kranevo, Bulgaria, June 19-25 2004, 277-282, ISBN: 9548986140

[AC3] F. Pambianco, A. A. Davydov, and S. Marcugini, Minimal 1-saturating sets and complete caps in binary projective geometries, Proceedings of ACCT 2004, Ninth International Workshop on Algebraic and Combinatorial Coding Theory, Kranevo, Bulgaria, June 19-25 2004, 113-119, ISBN: 9548986140

[AC2] F. Pambianco, S. Marcugini, and A. Milani, Classification of linear codes by preclassification, Proceedings of ACCT 2002, Eighth International Workshop on Algebraic and Combinatorial Coding Theory. Tsarskoe Selo, Russia, September 8-14, 2002, 200-203.

[AC1] F. Pambianco, S. Marcugini, and A. Milani, Existence and classification of NMDS codes over GF(5) and GF(7), Proceedings of ACCT 2000, Seventh International Workshop on Algebraic and Combinatorial Coding Theory, Bansko, Bulgaria, June 18-24, 2000, 232-239.

4. PhD THESIS

[TD] F. Pambianco, Classi di Weyl di varietà complesse, Tesi di Dottorato in Matematica, Università degli Studi “La Sapienza” di Roma, 1996.

[RT1] F. Pambianco, On Weyl classes on compact manifolds, Dipartimento di Matematica, Università degli Studi di Perugia, Rapporto Tecnico 2002-12.

CONTRIBUTED TALKS IN CONFERENCES

Combinatorics '94, Roma & Montesilvano (Pe), September 10-16, 1994. Small complete k-caps in PG(3,q), q odd.

Strutture Geometriche, Combinatoria, loro Applicazioni, Varenna (Co), 19-21 Aprile 1995. Archi e

calotte di cardinalità piccola in PG(n,q).

Combinatorics ’96, Assisi (Pg), September 8-14, 1996. On 10-arcs in projective planes.

GNSAGA , Perugia, 6-8 Novembre 1997. Una classe di calotte complete di uno spazio proiettivo sopra un campo di Galois. Membro del Comitato Organizzatore.

Combinatorics ’98, Mondello (Palermo), June 14-20, 1998. A 72-arc associated with the A_6-invariant sextic.

Strutture Geometriche, Combinatoria, loro Applicazioni, Caserta, 25-27 Febbraio, 1999. Rappresentazione ciclica di PG(n,q) e archi.

Combinatorics 2000, Gaeta, May 28 – June 3, 2000. The smallest size of a complete cap in PG(3,7).

ACCT 2000, Algebraic and Combinatorial Coding Theory, Bansko, Bulgaria, June 18-24, 2000. Existence and classification of NMDS codes over GF(5) and GF(7).

Discrete Mathematics and its Industrial Applications, Capri June 7-10, 2001. Different types of algorithms to search caps and codes.

Combinatorics 2002, Maratea, June 2-8, 2002. On maximal partial spreads in PG(3,9).

ACCT 2002, Algebraic and Combinatorial Coding Theory, San Pietroburgo, September 8-14, 2002. Classification of linear codes by preclassification.

Finite Geometries, first Irsee Conference, Irsee (Germany), February 16-22, 2003. Classification of codes exploiting an invariant.

Seventh International Conference on Finite Fields and Applications, May 5-9, 2003. Optimal Near MDS codes.

XV Convegno UMI, Milano, 8-13 Settembre 2003. Piani proiettivi, ricoprimenti e topologia di networks.

Convegno del Dipartimento di Matematica e Informatica (II), Perugia, 3-4 Ottobre 2003. Codici Near MDS: costruzione e classificazione.

ACCT 2004, Algebraic and Combinatorial Coding Theory, Kranevo, Bulgaria, September 19-25, 2004. AMDS codes of small dimension.

Combinatorics 2004, Acireale (CT), September 12-18, 2004. Minimal 1-saturating sets and complete caps in binary projective Geometries.

OC 2005, Fourth International Workshop on Optimal Codes and Related Topics, Pamporovo, Bulgaria, June 17-23 2005. Some highly symmetric codes.

Strutture Geometriche, Combinatoria e loro Applicazioni, Vicenza, 21-24 Settembre 2005. Codici di ricoprimento localmente ottimali e insiemi saturanti minimali.

Convegno del Dipartimento di Matematica e Informatica (IV), dedicato a Sauro Tulipani, Perugia, 13-14 Gennaio 2006. Codici di ricoprimento localmente ottimali e insiemi saturanti.

Combinatorics 2006, Ischia (Napoli), June 25th - July 1st, 2006. Complete (q^2 + q + 8)/2-caps in the projective caps in binary projective space PG(3,q), with odd prime q=2 (mod3).

ACCT 2006, Algebraic and Combinatorial Coding Theory, Zvenigorod, Russia, September 3-9, 2006. Complete caps in the projective space PG(3,q), with odd prime q; On the structure of the (n,r)-arcs in PG(2,q).

Combinatorics 2008, Costermano, June 22-28, 2008. Linear Covering codes over non binary finite fields.

ACCT 2008, Algebraic and Combinatorial Coding Theory, Pamporovo, Bulgaria, June 16-22, 2008. Semiovals in projective planes of small order, On the spectrum of sizes of complete caps in projective spaces PG(n,q) of small dimension.

Combinatorics 2010, Verbania, June 27 - July 3, 2010. Geometric constructions and upper bounds for complete arcs in PG(2,q).

ACCT 2010, Algebraic and Combinatorial Coding Theory, Akademgorodok, Novosibirsk, Russia, September 5-11, 2010. Symmetric arcs and curves, The spectrum of linear pure quantum [[n,n-10,4]]-codes.

XIX Congresso UMI, Bologna 12-17 settembre 2011. Archi in PG(2,q).

ACCT 2012, Algebraic and Combinatorial Coding Theory, Pomorie, Bulgaria, June 15-21, 2012. Multiple Coverings of the farthest off points and multiple saturating sets in projective spaces, The maximum and minimum sizes of complete (n,3)-arcs in PG(2,16).

Combinatorics 2012, Perugia, September 9-15, 2012. Multiple Coverings of the farthest off points and multiple saturating sets in projective spaces.

TEACHING EXPERIENCES

1992/93 – 1997/98Geometria and activities in the course of Algebra e Geometria, Facoltà di Ingegneria.

1996/97Matematica Generale, Corso di Laurea in Economia del Turismo e Corso di Diploma in Economia e Gestione dei Servizi Turistici, Università degli Studi di Perugia.

Matematica, Corso di Diploma in Gestione Tecnica ed Amministrativa in Agricoltura; orientamento: Gestione e Conservazione dell’Ambiente, Università degli Studi di Perugia.

1997/98

Matematica, Corso di Laurea in Scienze e Tecnologie della Produzione Animale, Università degli Studi di Perugia.

1998/99 - 1999/2000 Geometria, Corso di Laurea in Ingegneria Meccanica e in Ingegneria per l’Ambiente e il Territorio, Università degli Studi di Perugia.

From 2000/2001 to 2003/2004 Matematica A, Corso di Laurea Triennale in Ingegneria Meccanica, Università degli Studi di Perugia.

From 2004/2005 to 2006/2007Matematica A, Corso di Laurea Triennale in Ingegneria Meccanica, Università degli Studi di Perugia.

2003/2004Cycle of lessons inside the course Geometria Combinatoria II, Corso di Laurea Specialistica in Matematica, Università degli Studi di Perugia.

From 1999/2000 to 2009/2010Matematica, Corso di Laurea a Ciclo Unico in Chimica e Tecnologie Farmaceutiche, Facoltà di Farmacia, Università degli Studi di Perugia.

From 2003/2004 to 2008/2009Metodi Algebrici per l’Informazione – modulo II, Laurea Specialistica in Informatica e Telecomunicazioni, Università degli Studi di Perugia.

From 2009/2010 to 2010/2011- Metodi Algebrici per l’Informazione – modulo I, Laurea Magistrale in Informatica e Automazione, Università degli Studi di Perugia.- Metodi Matematici per l’Informazione – modulo I, titolarità per responsabilità didattica, Laurea Magistrale in Elettronica e Telecomunicazioni, Università degli Studi di Perugia.

From 2008/2009 to 2010/2011Geometria I - modulo B, Corso di Laurea Triennale in Ingegneria Informatica ed Elettronica, Università degli Studi di Perugia.

From 2011/2012 - todayGeometria I - modulo A, Corso di Laurea Triennale in Ingegneria Informatica ed Elettronica, Università degli Studi di Perugia.

From 2007/2008 - todayMatematica II - modulo I, Corso di Laurea Triennale in Ingegneria Meccanica, Università degli Studi di Perugia.

From 2010/2011 - today Matematica con Elementi di Informatica, Corso di Laurea a ciclo unico in Chimica e Tecnologie Farmaceutiche, Facoltà di Farmacia, Università degli Studi di Perugia.

She contributed as a reviewer to the composition of Degree and Phd thesis of students.