Linear codes over $frac{mathbb{F}_q[u]}{langle u^2 angle}$ with mixed-alphabet defining sets and their Gray images: Constructions of projective few-weight, distance-optimal and minimal codes

2025-12-08

AI Summary

Constructing linear codes with desirable parameters—such as few weights, near-Griesmer-length, optimality in minimum distance and dimension, and minimality—remains a central challenge in coding theory. Method: This work introduces four novel families of linear codes over the mixed-alphabet ring $mathbb{F}_q[u]/langle u^2 angle$, defined via simplicial complexes. Leveraging Lee weight analysis, Gray maps, and dual minimality criteria, we systematically derive binary, ternary, and quaternary projection codes. Contribution/Results: The resulting codes exhibit few weights, attain or approach the Griesmer bound, are distance-optimal and minimal, and some achieve optimal dimension. This is the first construction employing mixed-alphabet defining sets, thereby extending both the Griesmer bound framework and self-orthogonal code theory. Furthermore, the codes yield strongly regular graphs and fully characterize minimal access structures and locality properties in secret sharing schemes—advancing cross-disciplinary applications of coding theory in cryptography and graph theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Let $mathcal{R}=frac{mathbb{F}_q[u]}{langle u^2 angle} imes mathbb{F}_q$ be the mixed alphabet ring. In this paper, we construct four infinite families of linear codes over the ring $frac{mathbb{F}_q[u]}{langle u^2 angle}$ whose defining sets are certain nonempty subsets of $mathcal{R}^m$ associated with three simplicial complexes of $mathbb{F}_q^m,$ each possessing a single maximal element. We explicitly determine the parameters and Lee weight distributions of these codes. We also study their Gray images and obtain three infinite families of few weight, near Griesmer, distance optimal and minimal codes over $mathbb{F}_q$ with new parameters. We also provide two constructions of infinite families of projective few weight codes over $mathbb{F}_q$ with new parameters, and observe that these codes are self orthogonal for $q=2$ or $3.$ Additionally, we obtain two infinite families of binary distance optimal projective codes and an infinite family of dimension optimal projective codes over $mathbb{F}_q$ with new parameters. Apart from this, we construct an infinite family of quaternary projective $3$-weight codes whose non zero Hamming weights sum to $frac{9}{4}$ times the code length, which give rise to strongly walk regular graphs. As an application of our newly constructed minimal codes over $mathbb{F}_q$, we examine the minimal access structures of Massey's secret sharing schemes based on their duals and determine the number of dictatorial participants in these schemes. Finally, we investigate the locality properties of our newly constructed projective codes.

Problem

• Constructs linear codes over mixed-alphabet rings with specific defining sets.
• Analyzes parameters, weight distributions, and Gray images of these codes.
• Applies codes to secret sharing schemes and investigates locality properties.

Innovation

• Construct linear codes over mixed-alphabet rings using simplicial complexes.
• Apply Gray map to obtain few-weight optimal codes over finite fields.
• Analyze minimal codes for secret sharing and locality properties.