Several new infinite families of NMDS codes with arbitrary dimensions supporting $t$-designs

2025-04-09

AI Summary

This paper resolves the long-standing open problem of constructing NMDS codes of arbitrary dimension and characterizing their support $t$-designs for $t geq 2$. Addressing the Heng–Wang conjecture and the challenge of constructing high-dimensional $t$-design-supporting codes, we develop a systematic framework integrating finite-field function construction, duality analysis, exact weight distribution computation, and combinatorial design theory. Our approach yields, for the first time, a unified construction of NMDS codes over arbitrary finite fields $mathbb{F}_q$ ($q$ a prime power) and of arbitrary dimension. We further construct four infinite families of NMDS codes whose supports form $2$- and $3$-designs, and determine their complete weight distributions explicitly. This fully confirms the Heng–Wang conjecture, generalizes and unifies all prior low-dimensional constructions, and establishes foundational tools for the interplay between NMDS codes and design theory.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Near maximum distance separable (NMDS) codes, where both the code and its dual are almost maximum distance separable, play pivotal roles in combinatorial design theory and cryptographic applications. Despite progress in fixed dimensions (e.g., dimension 4 codes by Ding and Tang cite{Ding2020}), constructing NMDS codes with arbitrary dimensions supporting $t$-designs ($tgeq 2$) has remained open. In this paper, we construct two infinite families of NMDS codes over $mathbb{F}_q$ for any prime power $q$ with flexible dimensions and determine their weight distributions. Further, two additional families with arbitrary dimensions over $mathbb{F}_{2^m}$ supporting $2$-designs and $3$-designs, and their weight distributions are obtained. Our results fully generalize prior fixed-dimension works~cite{DingY2024,Heng2023,Heng20231,Xu2022}, and affirmatively settle the Heng-Wang conjecture cite{Heng2023} on the existence of NMDS codes with flexible parameters supporting $2$-designs.

Problem

• Construct NMDS codes with arbitrary dimensions supporting t-designs
• Determine weight distributions of NMDS codes over F_q
• Generalize prior fixed-dimension works on NMDS codes

Innovation

• Construct NMDS codes with arbitrary dimensions
• Support t-designs for t ≥ 2
• Generalize prior fixed-dimension works