A Characterization of $q$-Ary $2$-Designs and a $3$-Design Obstruction for Doubly-Extended Reed-Solomon Codes

Abstract

Let ${\mathcal{C}}_k$ be the doubly extended Reed--Solomon code with parameters $[q+1,k,q-k+2{]}_q$, where $q\ge 3$ is a prime power and $2\le k\le q$. We prove a conjecture of Wu and Zhou by showing that the minimum-weight codewords of ${\mathcal{C}}_k$ do not form a $q$-ary $3$-design when $2\le k\le q-1$. Using shortening, we extend this obstruction to every nonempty weight class with $3\le w<q+1$. We also establish that each individual nonempty positive-weight class of ${\mathcal{C}}_k$ forms a $q$-ary $2$-design if and only if $\gcd (k-1,q-1)=1$. Consequently, every nonempty positive-weight class below full support has exact $q$-ary design strength $2$ when this coprimality condition holds, and $1$ otherwise.The three-design obstruction follows by applying a classical subset-product character identity to a marginal covering count for three prescribed nonzero symbols. The comparison between weights rests on a covering-count formulation of the classical MDS coset-weight relations. For every linear $[n,k,d{]}_q$ MDS code with $2\le k\le n-1$, this formulation shows that the $q$-ary $2$-design property at any one nonempty positive weight is equivalent to the same property at every nonempty positive weight.More generally, for $2\le t\le \min \{k,d\}$, we prove that the $q$-ary $t$-design property at any $t-1$ distinct nonempty positive weights forces it at all nonempty positive weights.

arXiv comment18 pages