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

2026-03-13

AI Summary

This study investigates whether codewords of fixed weight in linear codes can support $q$-ary $t$-designs, addressing a notable gap in the theoretical literature. By establishing two general criteria and complementing them with automorphism group analysis to handle cases where the criteria fail, the work provides the first systematic method for determining when a linear code supports a $q$-ary design. Integrating tools from algebraic coding theory, combinatorial design, and group actions—and leveraging code operations such as duality, shortening, and puncturing—the paper demonstrates that specific weight classes of several classical codes, including extremal self-dual codes and extended Reed–Solomon codes, indeed form $q$-ary 2-designs with explicitly determined parameters, thereby significantly broadening the known constructions of $q$-ary designs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Game Theory and Economic Paradigms: Mechanism Design
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• linear codes
• q-ary t-designs
• combinatorial designs
• codeword weights
• design theory

Innovation

• q-ary t-designs
• linear codes
• design criteria
• automorphism group
• Reed-Solomon codes