2020/05/25 by Maria Bras-Amorós, Bras-Amorós, Maria, Alonso S. Castellanos +3
Computer Science · Mathematics · #11T71 #14G50 #14Q05 #94B27 #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.IT #math.AG #math.IT #msc:11T71 #msc:14G50 #msc:14Q05 #msc:94B27
paper · pdf · doi:10.48550/arxiv.2005.12239
To appear in IEEE Transactions on Information Theory
arxiv created 2021/11/03 · arxiv updated 2021/11/04
A flag of codes C0 \subsetneq C1 \subsetneq ⋯ \subsetneq Cs ⊆ \mathbb Fqn is said to satisfy the \it isometry-dual property if there exists \bf x∈ (\mathbbFq^*)n such that the code Ci is \bf x-isometric to the dual code Cs-i^⊥ for all i=0,…, s. For P and Q rational places in a function field \mathcal F, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes C_\mathcal L(D, a0P+bQ)\subsetneq C_\mathcal L(D, a1P+bQ)\subsetneq … \subsetneq C_\mathcal L(D, asP+bQ), where the divisor D is the sum of pairwise different rational places of \mathcal F and P, Q are not in supp(D). We characterize those sequences in terms of b for general function fields. We then apply the result to the broad class of Kummer extensions \mathcal F defined by affine equations of the form ym=f(x), for f(x) a separable polynomial of degree r, where gcd(r, m)=1. For P the rational place at infinity and Q the rational place associated to one of the roots of f(x), it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if m divides 2b+1. At the end we illustrate our results by applying them to two-point codes over several well know function fields.