Research on Linear Codes Holding $q$-Ary $t$-Designs

A $q$-ary $t$-$(n,w,\lambda)$ design is a collection $\mathcal{A}$ of vectors of weight $w$ in $\mathbb{F}_{q}^{n}$ with the property that every vector of weight $t$ in $\mathbb{F}_{q}^{n}$ is contained in exactly $\lambda$ members of $\mathcal{A}$. The supports of the vectors in a $q$-ary $t$-design form an ordinary $t$-design, possibly with repeated blocks. While linear codes supporting ordinary combinatorial designs have been extensively studied, the case where codes hold $q$-ary designs remains largely unexplored. This motivates a systematic investigation into whether codewords of a fixed weight in a linear code can form a $q$-ary $t$-design. Building on previous work, we develop two new criteria for this purpose. Applying these criteria, we show that several families of linear codes hold $q$-ary $2$-designs, including one- and two-weight codes, extremal self-dual codes, as well as certain dual codes, shortened codes, and punctured codes derived from them. Moreover, for linear codes that do not satisfy these criteria, we provide an alternative approach based on the automorphism group of the code. This method enables the construction of $q$-ary $2$-designs from doubly-extended Reed-Solomon codes. Notably, for a class of linear codes previously known to support $4$-designs, we demonstrate that their codewords of certain weights give rise to $q$-ary $2$-designs whose parameters are precisely determined.