The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes
share
A flag of codes C_0 ⊊ C_1 ⊊...⊊ C_s ⊆F_q^n is said to satisfy the isometry-dual property if there exists x∈ (F_q^*)^n such that the code C_i is x-isometric to the dual code C_s-i^ for all i=0,..., s. For P and Q rational places in a function field F, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes C_ L(D, a_0P+bQ)⊊ C_ L(D, a_1P+bQ)⊊...⊊ C_ L(D, a_sP+bQ), where the divisor D is the sum of pairwise different rational places of F and P, Q are not in (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 F defined by affine equations of the form y^m=f(x), for f(x) a separable polynomial of degree r, where (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.
READ FULL TEXT
