The Isometry-Dual Property in Flags of Many-Point Algebraic Geometry Codes
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Let π½_q be the finite field with q elements and let β be the set of non-negative integers. A flag of linear codes C_0 β C_1 ββ―β C_s is said to have the isometry-dual property if there exists a vector xβ (π½_q^*)^n such that C_i= xΒ· C_s-i^β₯, where C_i^β₯ denotes the dual code of the code C_i. Consider β± a function field over π½_q, and let P and Q_1,β¦, Q_t be rational places in β±. Let the divisor D be the sum of pairwise different places of β± such that P, Q_1,β¦, Q_t are not in (D), and let G_Ξ² be the divisor β_i=1^tΞ²_iQ_i, for given Ξ²_i's ββ€. For suitable values of Ξ²_i's in β€ and varying an integer a we investigate the existence of isometry-dual flags of codes in the families of many-point algebraic geometry codes C_β(D, a_0P+ G_Ξ²)β C_β(D, a_1P+ G_Ξ²))ββ¦β C_β(D, a_sP+ G_Ξ²)). We then apply the obtained results to the broad class of Kummer extensions β± defined by affine equations of the form y^m=f(x), for f(x) a separable polynomial of degree r, where (r, m)=1. In particular, depending on the place P, we obtain necessary and sufficient conditions depending on m and Ξ²_i's such that the flag has the isometry-dual property.
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