$\mathbb{F}_q$-零的稀疏三元多项式和环面三叠码

IF 1.6 2区 数学 Q2 MATHEMATICS, APPLIED
Kyle P. Meyer, Ivan Soprunov, Jenya Soprunova
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引用次数: 0

摘要

对于给定的晶格多面体P∧R,考虑有限域Fq上三元多项式的空间LP,其牛顿多面体包含在P中。我们根据P和q的Minkowski长度(域的大小)给出了LP中多项式的fq - 0的最大数目的上界。因此,这产生了通过计算代数环面(Fq)点上的LP元素来定义的环码最小距离的下界。我们的方法是基于对LP中尽可能多的非单位因子的多项式分解的理解。我们得到的相关组合结果是P中包含有最大可能数目的非平凡和的格多边形的闵可夫斯基和的描述。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
$\mathbb{F}_q$-Zeros of Sparse Trivariate Polynomials and Toric 3-Fold Codes
For a given lattice polytope P ⊂ R, consider the space LP of trivariate polynomials over a finite field Fq, whose Newton polytopes are contained in P . We give upper bounds for the maximum number of Fq-zeros of polynomials in LP in terms of the Minkowski length of P and q, the size of the field. Consequently, this produces lower bounds for the minimum distance of toric codes defined by evaluating elements of LP at the points of the algebraic torus (Fq). Our approach is based on understanding factorizations of polynomials in LP with the largest possible number of non-unit factors. The related combinatorial result that we obtain is a description of Minkowski sums of lattice polytopes contained in P with the largest possible number of non-trivial summands.
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CiteScore
2.20
自引率
0.00%
发文量
19
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