The Isometry-Dual Property in Flags of Many-Point Algebraic Geometry Codes

06/10/2021

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by   Maria Bras-Amorós, et al.

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Let 𝔽_q be the finite field with q elements and let ℕ be the set of non-negative integers. A flag of linear codes C_0 ⊊ C_1 ⊊⋯⊊ C_s is said to have the isometry-dual property if there exists a vector x∈ (𝔽_q^*)^n such that C_i= x· C_s-i^⊥, where C_i^⊥ denotes the dual code of the code C_i. Consider ℱ a function field over 𝔽_q, and let P and Q_1,…, Q_t be rational places in ℱ. Let the divisor D be the sum of pairwise different places of ℱ such that P, Q_1,…, Q_t are not in (D), and let G_β be the divisor ∑_i=1^tβ_iQ_i, for given β_i's ∈ℤ. For suitable values of β_i's in ℤ 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_ℒ(D, a_0P+ G_β)⊊ C_ℒ(D, a_1P+ G_β))⊊…⊊ C_ℒ(D, a_sP+ G_β)). We then apply the obtained results 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. In particular, depending on the place P, we obtain necessary and sufficient conditions depending on m and β_i's such that the flag has the isometry-dual property.