Papers by Massimo Giulietti
Inspired by a recent paper of Garcia, Stichtenoth and Xing [2000, Compositio Math. 120, 137--170]... more Inspired by a recent paper of Garcia, Stichtenoth and Xing [2000, Compositio Math. 120, 137--170], we investigate the quotient curves of the Deligne-Lusztig curve associated to the Suzuki group.
Bicovering arcs in Galois affine planes of odd order are a powerful tool for constructing complet... more Bicovering arcs in Galois affine planes of odd order are a powerful tool for constructing complete caps in spaces of higher dimensions. In this paper we investigate whether some arcs contained in nodal cubic curves are bicovering. For $m_1$, $m_2$ coprime divisors of $q-1$, bicovering arcs in $AG(2,q)$ of size $k\le (q-1)\frac{m_1+m_2}{m_1m_2}$ are obtained, provided that $(m_1m_2,6)=1$ and $m_1m_2<\sqrt[4]{q}/3.5$. Such arcs produce complete caps of size $kq^{(N-2)/2}$ in affine spaces of dimension $N\equiv 0 \pmod 4$. For infinitely many $q$'s these caps are the smallest known complete caps in $AG(N,q)$, $N \equiv 0 \pmod 4$.
Complete arcs and complete caps from cubics with an isolated double point
Small complete arcs and caps in Galois spaces over finite fields $\fq$ with characteristic greate... more Small complete arcs and caps in Galois spaces over finite fields $\fq$ with characteristic greater than 3 are constructed from cubic curves with an isolated double point. For $m$ a divisor of $q+1$, complete plane arcs of size approximately $q/m$ are obtained, provided that $(m,6)=1$ and $m<\{1}{4}q^{1/4}$. If in addition $m=m_1m_2$ with $(m_1,m_2)=1$, then complete caps of size approximately $\{m_1+m_2}{m}q^{N/2}$ in affine spaces of dimension $N\equiv 0 \pmod 4$ are constructed.
Decoding Goppa Codes with MAGMA
Ars Combinatoria, 2001
ABSTRACT
We give a geometric interpretation of additive quantum stabilizer codes in terms of sets of lines... more We give a geometric interpretation of additive quantum stabilizer codes in terms of sets of lines in binary symplectic space. It is used to obtain synthetic geometric constructions and non-existence results. In particular several open problems are removed from Grassl's database (13).
Designs, Codes and Cryptography, 2015
Complete (k, 3)-arcs in projective planes over finite fields are the geometric counterpart of lin... more Complete (k, 3)-arcs in projective planes over finite fields are the geometric counterpart of linear non-extendible Near MDS codes of length k and dimension 3. A class of infinite families of complete (k, 3)-arcs in PG(2, q) is constructed, for q a power of an odd prime p ≡ 2 (mod 3). The order of magnitude of k is smaller than q. This property significantly distinguishes the complete (k, 3)-arcs of this paper from the previously known infinite families, whose size differs from q by at most 2 √ q.
IEEE Transactions on Information Theory, 2015
For a linear maximum distance separable (MDS) code with redundancy r, the covering radius is eith... more For a linear maximum distance separable (MDS) code with redundancy r, the covering radius is either r or r − 1. However, for r > 3, few examples of q-ary linear MDS codes with radius r − 1 are known, including the Reed-Solomon codes with length q + 1. In this paper, for redundancies r as large as 12 √ q, infinite families of q-ary MDS codes with covering radius r − 1 and length less than q + 1 are constructed. These codes are obtained from algebraic-geometric codes arising from elliptic curves. For most pairs (r, q) with r ≤ 12 √ q, these are the shortest q-ary MDS codes with covering radius r − 1.
Designs Codes and Cryptography, 2002
We show that the set of F q -rational points of either certain Fermat curves or certain F q -Frob... more We show that the set of F q -rational points of either certain Fermat curves or certain F q -Frobenius non-classical plane curves is a complete (k, d)-arc in P 2 (F q ), where k and d are respectively the number of F q -rational points and the degree of the underlying curve.
ON THE NUMBER OF RATIONAL POINTS OF GENERALIZED FERMAT CURVES OVER FINITE FIELDS
International Journal of Number Theory, 2012
ABSTRACT The authors define a generalized Fermat curve to be a curve defined over 𝔽 q that satisf... more ABSTRACT The authors define a generalized Fermat curve to be a curve defined over 𝔽 q that satisfies the following properties: (i) the 𝔽 q -automorphism group of the curve contains the direct product G of two cyclic groups C 1 and C 2 of order k prime to p and greater than 2, (ii) the quotient curve with respect to either C 1 or C 2 is rational, (iii) each short orbit under the action of G is preserved by the 𝔽 q -Frobenius morphism. In the article under review, the authors obtain new upper bounds for the number of rational points of a generalized Fermat curve. This generalizes some previous results on Fermat curves.
Bicovering arcs and small complete caps from elliptic curves
Journal of Algebraic Combinatorics, 2013
Small complete caps from singular cubics, II
On Dense Sets Related To Plane Algebraic Curves
Ars Combinatoria - ARSCOM, 2004
We show that certain subsets of Fq-rational points of the curve XZn 1 = Y n are dense sets in P2(... more We show that certain subsets of Fq-rational points of the curve XZn 1 = Y n are dense sets in P2(Fq).
Mathematische Annalen, 2009
A new family of F q 2 -maximal curves is presented and some of their properties are investigated.
Journal of the London Mathematical Society, 2010
In positive characteristic, algebraic curves can have many more automorphisms than expected from ... more In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g 4 automorphisms. It has been observed on many occasions that the most anomalous examples invariably have zero p-rank. In this paper, the K-automorphism group Aut(X ) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that if the curve has genus g ≥ 2 and |Aut(X )| > 24g 2 , then Aut(X ) has a fixed point on X , apart from few exceptions. In the exceptional cases the possibilities for Aut(X ) and g are determined.
Journal of Combinatorial Designs, 2009
Some new families of small complete caps in PG(N, q), q even, are described. By using inductive a... more Some new families of small complete caps in PG(N, q), q even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N, q), N ≥ 4, for all q ≥ 2 3 .I np a r t i c u l a r , substantial improvements are obtained for infinite values of q square, including q = 2 2Cm , C ≥ 5, m ≥ 3;f o rq = 2 Cm , C ≥ 5, m ≥ 9,w i t hC, m odd; and for all q ≤ 2 18 . q 2009 Wiley Periodicals, Inc. J Combin Designs
Small Complete Caps from Singular Cubics
Journal of Combinatorial Designs, 2013
ABSTRACT Small complete arcs and caps in Galois spaces over finite fields F-q with characteristic... more ABSTRACT Small complete arcs and caps in Galois spaces over finite fields F-q with characteristic greater than three are constructed from singular cubic curves. For m a divisor of q + 1 or q - 1, complete plane arcs of size approximately q/m are obtained, provided that (m, 6) = 1 and m &lt; 1/4q(1/4). If in addition m = m(1)m(2) with (m(1), m(2)) = 1, then complete caps in affine spaces of dimension N equivalent to 0 (mod 4) with roughly m(1)+m(2)/m q(N/2) points are described. These results substantially widen the spectrum of qs for which complete arcs in AG(2, q) of size approximately q(3/4) can be constructed. Complete caps in AG(N, q) with roughly q((4N-1)/8) points are also provided. For infinitely many qs, these caps are the smallest known complete caps in AG(N, q), N equivalent to 0 (mod 4).
Journal of Algebraic Combinatorics, 2013
A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive ... more A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and they play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.
IEEE Transactions on Information Theory, 2000
Some new infinite families of short quasi-perfect linear codes are described. Such codes provide ... more Some new infinite families of short quasi-perfect linear codes are described. Such codes provide improvements on the currently known upper bounds on the minimal length of a quasi-perfect [n; n 0m; 4] -code when either 1) q = 16; m 5; m odd, or 2) q = 2 ; 7 i 15; m 4, or 3) q = 2 ;` 8; m 5; m odd. As quasi-perfect [n; n0m; 4] -codes and complete n-caps in projective spaces P G(m 01;q) are equivalent objects, new upper bounds on the size of the smallest complete cap in P G(m 01;q) are obtained.
IEEE Transactions on Information Theory, 2000
In this paper, algebraic-geometric (AG) codes associated to a recently discovered class of maxima... more In this paper, algebraic-geometric (AG) codes associated to a recently discovered class of maximal curves are investigated. As a result, some linear codes with better parameters with respect to the previously known ones are discovered, and 70 improvements on MinT's tables (''Tables of optimal parameters for linear codes,'' University of Salzburg, Salzburg, Austria) are obtained.
Graphs and Combinatorics, 2013
We propose geometrical methods for constructing square 01-matrices with the same number n of unit... more We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m, n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.
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Papers by Massimo Giulietti