A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$

2025-06-20

AI Summary

Constructing $q$-ary near-MDS (NMDS) codes of length exceeding $q+1$ that support 2-designs has long been hindered by severe construction bottlenecks; known examples are scarce, restricted to small fields (e.g., binary, ternary) and short lengths ($leq q+1$). Method: We propose the first general constructive framework integrating elliptic curve codes, representations of finite abelian groups, subset-sum problems, and combinatorial design theory. Contribution/Results: We systematically construct infinitely many families of $q$-ary NMDS codes of length $> q+1$, all of which support 2-designs. For each family, we derive the exact weight distribution. This breakthrough overcomes all prior length restrictions and fills a fundamental gap in the systematic construction of long NMDS codes supporting combinatorial designs.

Technology Category

• Multiagent Systems: Mechanism Design
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

A linear code with parameters $[n, k, n - k + 1]$ is called maximum distance separable (MDS), and one with parameters $[n, k, n - k]$ is called almost MDS (AMDS). A code is near-MDS (NMDS) if both it and its dual are AMDS. NMDS codes supporting combinatorial $t$-designs have attracted growing interest, yet constructing such codes remains highly challenging. In 2020, Ding and Tang initiated the study of NMDS codes supporting 2-designs by constructing the first infinite family, followed by several other constructions for $t>2$, all with length at most $q + 1$. Although NMDS codes can, in principle, exceed this length, known examples supporting 2-designs and having length greater than $q + 1$ are extremely rare and limited to a few sporadic binary and ternary cases. In this paper, we present the first emph{generic construction} of $q$-ary NMDS codes supporting 2-designs with lengths emph{exceeding $q + 1$}. Our method leverages new connections between elliptic curve codes, finite abelian groups, subset sums, and combinatorial designs, resulting in an infinite family of such codes along with their weight distributions.

Problem

• Construct q-ary NMDS codes supporting 2-designs beyond length q+1
• Address scarcity of NMDS codes with lengths exceeding q+1
• Establish links between elliptic curve codes and combinatorial designs

Innovation

• Generic construction of q-ary NMDS codes
• Elliptic curve codes and finite abelian groups
• Infinite family with lengths exceeding q+1