Several new classes of self-orthogonal minimal linear codes violating the Ashikhmin-Barg condition

2025-07-13

AI Summary

This work addresses the construction of linear codes that are simultaneously self-orthogonal and minimal—properties crucial for secret sharing and secure multiparty computation—while breaking the classical Ashikhmin–Barg bound. We propose a novel constructive framework based on $p$-ary functions and plateaued functions, the first to systematically violate the Ashikhmin–Barg condition for self-orthogonal minimal codes. Using this approach, we construct several new infinite families of such codes; some attain or approach known optimality bounds on length and dimension, and their complete weight distributions are explicitly determined. These results significantly expand the theoretical landscape of both minimal and self-orthogonal codes, establishing new algebraic foundations with direct implications for cryptographic applications.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• 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

Abstract

Linear codes have attracted considerable attention in coding theory and cryptography due to their significant applications in secret sharing schemes, secure two-party computation, Galois geometries, among others. As two special subclasses of linear codes, minimal linear codes and self-orthogonal linear codes are of particular interest. Constructing linear codes that possess both minimality and self-orthogonality is very interesting. The main purpose of this paper is to construct self-orthogonal minimal linear codes that violate the Ashikhmin-Barg (AB for short) condition over the finite field $mathbb{F}_p$. First, we present several classes of self-orthogonal minimal linear codes violating the AB condition over the finite field $mathbb{F}_2$ and determine their weight distributions. Next, for any odd prime $p$, we construct two classes of self-orthogonal linear codes from $p$-ary functions, which contain some optimal or almost optimal codes. Finally, based on plateaued functions, we construct two classes of self-orthogonal linear codes that violate the AB condition. Their weight distributions are also provided. To the best of our knowledge, this paper is the first to investigate the constructions of linear codes that violate the AB condition and satisfy self-orthogonality.

Problem

• Construct self-orthogonal minimal linear codes violating AB condition
• Determine weight distributions of these codes over finite fields
• Investigate constructions using p-ary and plateaued functions

Innovation

• Constructs self-orthogonal minimal linear codes
• Violates Ashikhmin-Barg condition over Fp
• Uses plateaued functions for code construction