最小舒伯特码码字与次最小格拉斯曼码码字
Minimum Schubert Codewords and Second-Minimum Grassmann Codewords
- IIT Jammu（印度技术学院詹穆分校）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文解决了舒伯特码最小权码字分类的猜想，据此刻画了格拉斯曼码次最小权码字的充要条件，并完成了其计数。
中文摘要 AI 辅助
本文对舒伯特码$C_\alpha(\ell, m)$的最小权码字进行分类，解决了Ghorpade与Singh在2018年提出的猜想，该分类适用于所有$q$值与所有$\alpha$。我们利用该分类证明，格拉斯曼码$C(\ell, m)$的码字为次最小权码字当且仅当它由$\bigwedge^{m-\ell}V$中可表示为可分解的$(m-\ell-2)$-向量与秩为4的交错2-向量乘积的元素索引。最后，我们给出了格拉斯曼码次最小权码字的计数结果。
英文摘要
In this paper, we give a classification of the minimum weight codewords of Schubert codes $C_α(\ell, m)$ by settling the conjecture proposed by Ghorpade and Singh in 2018, for all values of $q$ and all $α$. We use this classification to prove that a codeword of the Grassmann code $C(\ell, m)$ has the second minimum weight if and only if it is indexed by an element of $\bigwedge^{m-\ell}V$ that can be written as the product of a decomposable $(m-\ell-2)$-vector and an alternating $2$-vector of rank $4$. Finally, we give an enumeration of the second minimum weight codewords of the Grassmann code.
