A $q$-ary $t$-$(n,w,λ)$ 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 $λ$ 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.

翻译：一个$q$元$t$-$(n,w,λ)$设计是$\mathbb{F}_{q}^{n}$中一组重量为$w$的向量集合$\mathcal{A}$，其满足性质：$\mathbb{F}_{q}^{n}$中每一个重量为$t$的向量都恰好包含在$\mathcal{A}$的$λ$个成员中。$q$元$t$设计中向量的支撑集构成一个通常的$t$设计，其中可能包含重复区组。虽然支撑通常组合设计的线性码已被广泛研究，但码支撑$q$元设计的情形在很大程度上仍未得到探索。这促使我们系统性地研究线性码中固定重量的码字是否能构成$q$元$t$设计。基于先前工作，我们为此目的提出了两个新的判定准则。应用这些准则，我们证明多类线性码支撑$q$元$2$设计，包括单重码和双重码、极值自对偶码，以及由它们衍生的某些对偶码、缩短码和穿孔码。此外，对于不满足这些准则的线性码，我们提供了一种基于码自同构群的替代方法。该方法使得从双扩展Reed-Solomon码构造$q$元$2$设计成为可能。值得注意的是，对于一类先前已知支撑$4$设计的线性码，我们证明了其特定重量的码字可产生参数完全确定的$q$元$2$设计。