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- W4287121268 abstract "Let $F_q$ be a finite field. A flag of $F_q$-linear codes $C_0subsetneq C_1subsetneqdotssubsetneq C_s$ is said to satisfy the isometry-dual property if there exists a vector $xin(F_q^*)^n$ such that $C_i=xcdot C_{s-i}^perp$, where $C_i^perp$ denotes the dual code of $C_i$. Consider $F/F_q$ a function field and let $P$ and $Q_1,ldots,Q_t$ be rational places of $F$. Let the divisor $D$ be the sum of pairwise different places of $F$ such that $P, Q_1,dots,Q_t$ are not in $supp(D)$. In a previous work we investigated the existence of flags of two-point codes $C(D,a_0P+bQ_1)subsetneq C(D,a_1P+bQ_1))subsetneqdotssubsetneq C(D,a_sP+bQ_1)$ satisfying the isometry-dual property for a non-negative integer $b$ and an increasing sequence of positive integers $a_0,dots,a_s$. While for one-point codes (i.e. for $b=0$) there is only need to analyze positive integers $a$, for the case of $(t+1)$-point codes, the integers $a$ may be negative. We extend our previous results in different directions. On one hand to the case of negative integers $a$ and $b$, and on the other hand we extend our results to flags of $(t+1)$-point codes $C(D,a_0P+sum_{i=1}^tbeta_iQ_i)subsetneq C(D, a_1P+sum_{i=1}^tbeta_iQ_i))subsetneqdotssubsetneq C(D, a_sP+sum_{i=1}^tbeta_iQ_i)$ for any tuple of (either positive or negative) integers $beta_1,dots,beta_t$ and for an increasing sequence of (either positive or negative) integers $a_0,dots,a_s$. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $gcd(r, m)=1$. In particular, depending on the place $P$ and for $D$ an $Aut(F_q(x, y)/F_q(x))$-invariant sum of rational places of $F$ such that $P,Q_inotin supp(D)$, we obtain necessary and sufficient conditions on $m$ and $beta_i$'s such that the flag has the isometry-dual property." @default.
- W4287121268 created "2022-07-25" @default.
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- W4287121268 date "2021-06-10" @default.
- W4287121268 modified "2023-10-18" @default.
- W4287121268 title "Isometry-Dual Flags of Many-Point AG Codes" @default.
- W4287121268 doi "https://doi.org/10.48550/arxiv.2106.05600" @default.
- W4287121268 hasPublicationYear "2021" @default.
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