Bounding the Trellis State Complexity of Algebraic Geometric Codes

Abstract

Abstract. Let C be an algebraic geometric code of dimension k and length n constructed on a curve X over F q . Let s(C) be the state complexity of C and set w(C) := min{k, n−k}, the Wolf upper bound on s(C). We introduce a numerical function R that depends on the gonality sequence of X and show that s(C) ≥ w(C) − R(2g − 2), where g is the genus of X . As a matter of fact, R(2g − 2) ≤ g − (γ 2 − 2) with γ 2 being the gonality over F q of X , and thus in particular we have that s(C) ≥ w(C) − g + γ 2 − 2.

Introduction to algebraic geometry codes

“…Conversely, from Riemann-Roch theorem it follows that γ i ≤ i − 1 + g with equality for i > g. The gonality sequence GS(X ) verifies a symmetry property (similar to the symmetry property for semigroups): for every integer r, it holds that r ∈ GS(X ) if and only if 2g − 1 − r ∈ GS(X ), cf. [37]. In general, computing GS(X ) is a difficult task but for plane curves this sequence is entirely known and depends only on the degree of X , see [43].…”

Pesos de Hamming de códigos Castillo

“…Esta cota mejora la cota de 56,54,52,51,48,46,44,43,42,41,40,39,38,36,35,34,33,32,31,30,29,28,28,26,25,24,23,22,21,20,21,18,19,16,17,16,13,12,14,10,13,8,12,10,9,8,8,6,8,7,4,5,…”

Sobre codigos hermitianos generalizados

“…• Calcular pesos hierárquicos dos códigos GH s e refinar as cotas encontradas usando a generalização da distância Feng-Rao [22], [1], e complexidade de treliças [23].…”