← 研踪 PaperTrailarXiv:2611.05861 · v1
A Complete Characterization of $q$-Ary $2$-Designs in Doubly-Extended Reed-Solomon Codes
首次提交 2026-11-04 10:52
版本更新 2026-11-04 10:52
雷达观测 2026-11-05 10:08
摘要
Let $\mathcal{C}_k$ be the doubly-extended Reed--Solomon code with parameters $[q+1, k, q-k+2]_q$. We give a complete characterization of when its codewords of a fixed weight form $q$-ary 2-designs. The main ingredient is an exact criterion showing that the number of $r$-subsets of $\mathbb{F}_q^*$ with a prescribed product is independent of that product if and only if $\gcd(r, q-1)=1$. Applying this criterion to the minimum-weight codewords of $\mathcal{C}_k$, we prove that they form a $q$-ary 2-design if and only if $\gcd(k-1, q-1)=1$. We further establish a propagation result for MDS codes showing that the $q$-ary 2-design property at minimum weight implies the same property at every nonzero weight. Consequently, $\mathcal{C}_k$ holds $q$-ary 2-designs at all nonzero weights if and only if $\gcd(k-1, q-1)=1$.
作者备注：6 pages
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v1 · 2026-11-04 10:52 · 最新观测版本
A Complete Characterization of $q$-Ary $2$-Designs in Doubly-Extended Reed-Solomon Codes
Let $\mathcal{C}_k$ be the doubly-extended Reed--Solomon code with parameters $[q+1, k, q-k+2]_q$. We give a complete characterization of when its codewords of a fixed weight form $q$-ary 2-designs. The main ingredient is an exact criterion showing that the number of $r$-subsets of $\mathbb{F}_q^*$ with a prescribed product is independent of that product if and only if $\gcd(r, q-1)=1$. Applying this criterion to the minimum-weight codewords of $\mathcal{C}_k$, we prove that they form a $q$-ary 2-design if and only if $\gcd(k-1, q-1)=1$. We further establish a propagation result for MDS codes showing that the $q$-ary 2-design property at minimum weight implies the same property at every nonzero weight. Consequently, $\mathcal{C}_k$ holds $q$-ary 2-designs at all nonzero weights if and only if $\gcd(k-1, q-1)=1$.
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