Given an ordered set $B$ of a finite field, a combinatorial simplex over $B$ is defined as the set of vectors such that the positions of the entries, with respect to $B$, sum up to a fixed integer. CAP codes are Reed-Muller type codes defined over a combinatorial simplex. They were recently introduced by Kopparty et al. as a high-rate alternative to classical Reed-Muller codes, capable of achieving arbitrarily high rates close to one for any fixed minimum distance. In this paper, we use tools from commutative algebra to analyze a combinatorial simplex and its associated CAP code. We give a universal Gröbner basis for the vanishing ideal of a combinatorial simplex. We describe the generalized Hamming weights of a CAP code in terms of the footprint of the vanishing ideal. For the minimum distance case, we proved a closed formula. We give a set of polynomials whose evaluations on the combinatorial simplex generate the dual of the CAP code. We describe the affine permutations that leave invariant a combinatorial simplex and use this information to prove that, in some cases, the permutation group of a CAP code is a symmetric group.

翻译：给定有限域上的有序集$B$，定义组合单纯形为向量集合，其中各分量位置相对于$B$的和为固定整数。CAP码是定义在组合单纯形上的Reed-Muller类型码，由Kopparty等人最近提出，作为经典Reed-Muller码的高码率替代方案，能在任意固定最小距离下实现趋近于1的任意高码率。本文利用交换代数工具分析组合单纯形及其关联的CAP码。我们给出组合单纯形零化理想的通用Gröbner基；通过零化理想的足迹描述CAP码的广义汉明重量；针对最小距离情形，推导出闭式公式；给出多项式集合，其在组合单纯形上的评估值生成CAP码的对偶码；刻画保持组合单纯形不变的仿射置换，并利用此信息证明在某些情况下CAP码的置换群为对称群。