Conference Agenda

The problem of finding good codes is central to the theory of error correcting codes. For many years coding theorists have addressed this problem by adding algebraic and combinatorial structure to C.

In the early 80s Goppa used algebraic curves to construct linear error correcting codes, the socalled algebraic geometric codes (AG codes). The construction of an AG code with alphabet a finite field Fq requires that the underlying curve is Fq-rational and involves two Fq-rational divisors D and G on the curve.

In this minisymposium we will present results on Algebraic Geometry codes and their performances.

This talk presents results about Goppa codes over minimal Hirzebruch surfaces. Hirzebruch surfaces being toric surfaces, they are endowed with a polynomial coordinate ring, named the Cox ring. They also have a pleasant description as quotient spaces. These features enable us to define Goppa codes via the evaluation of polynomials similarly to projective Reed-Muller codes. Beside an easy implementation, considering polynomials makes us benefit from algebraic tools, such as Gröbner basis, to handle the parameters of these codes. Explicit formula for the dimension and the minimum distance of a Goppa code associated to a divisor D are computed as functions of the Picard class of D. The parameters are given for any size of the alphabet, even when the evaluation map is not injective. The minimum distance thus provides an upper bound of the number of rational points of a non-filling curve on a Hirzebruch surface. Moreover, the geometry of Hirzebruch surfaces, notably their ruling, leads to nice local decoding properties for these codes.

Hermitian functional and differential codes are AG-codes defined on a Hermitian curve. To ensure good performance, the divisors defining such AG-codes have to be carefully chosen, exploiting the rich combinatorial and algebraic properties of the Hermitian curves. In this paper, the case of differential codes C_Ω(D,mT) on the Hermitian curve H_q^3 defined over F_q^6 is worked out, where supp(T):=H_q^3(F_q^2), the set of all F_q^2-rational points of H_q^3, while D is taken, as usual, to be the sum of the points in the complementary set D = H_q^3(F_q^6}) H_q(F_q^2). For certain values of m, such codes C_Ω(D,mT) have better minimum distance compared with true values of 1-point Hermitian codes. The automorphism group of C_L(D,mT), m≤q^3-2, is isomorphic to PGU(3,q).

Let X be an algebraic curve defined over the finite field of order q. The parameters of the AG codes associated with X strictly depend on the underlying curve X. In general, curves with many rational places with respect to their genus give rise to AG codes with good parameters. For this reason maximal curves, that are curves attaining the Hasse-Weil upper bound, have been widely investigated in the literature. In this work, we focus our attention on the GK curve, which is a maximal curve constructed by Giulietti and Korchmáros which cannot be covered by the Hermitian curve whenever q is odd. In particular we investigate the minimum distance and the weight distribution of dual AG codes arising from the Giulietti-Korchmáros maximal curves. In most cases, the weight distribution of a given code is hard to be computed. Even the problem of computing codewords of minimum weight can be a difficult task, apart from specific cases. We do so using the link between the weight of the codewords of such codes and the geometry of the curve. We compute the maximal number of intersections that the GK curve can have with plane curves of low degree and we use this fact to determine the actual minimum distance and the number of minimum weight codewords of dual one-point AG codes arising from the GK curve.

In this talk, we will discuss explicit subcovers of a class of trace-defining curves over a finite field. It turns out that all such subcovers have a distiguished rational point P, for which the Weierstrass semigroup H(P) can often be determined. This will lead us to the construction of the corresponding one-point AG codes with very good parameters. In particular, we will present improvements on the parameters of at least 108 codes from the MinT table.