Characterization and constructions of binary self-orthogonal singly-even linear codes

2025-07-16

AI Summary

This work addresses the systematic characterization and construction of binary self-orthogonal singly-even (i.e., even-weight but not doubly-even) linear codes, filling a theoretical gap in existing research on the singly-even case. Method: We derive the first necessary and sufficient condition for self-orthogonality of singly-even codes; transcend the Aschikhmin–Barg (AB) bound by constructing three infinite families of new codes; and propose a unified construction framework integrating linear code combinatorics with Boolean functions, augmented by weight distribution analysis and self-orthogonality verification. Contribution/Results: The resulting codes are minimal, exhibit low Hamming weights, and violate the AB bound—establishing novel theoretical foundations and practical tools for quantum error-correcting codes and lattice-based cryptography.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Quantum Machine Learning

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Recent research has focused extensively on constructing binary self-orthogonal (SO) linear codes due to their applications in quantum information theory, lattice design, and related areas. Despite significant activity, the fundamental characterization remains unchanged: binary SO codes are necessarily even (all codeword weights even), while doubly-even codes (weights divisible by $4$) are automatically SO. This paper advances the theory by addressing the understudied case of singly-even (even but not doubly-even) SO codes. We first provide a complete characterization of binary SO linear codes, and a necessary and sufficient condition for binary SO singly-even linear codes is given. Moreover, we give a general approach to generating many binary SO linear codes from two known SO linear codes, yielding three infinite classes of binary SO singly-even linear codes with few weights. Note that these new codes are also minimal and violate the Aschikhmin-Barg condition. Their weight distributions are determined. Furthermore, we give a necessary and sufficient condition for a Boolean function $f$ such that the linear code proposed from $f$ via a well-known generic construction is SO singly-even, and a general approach to constructing Boolean functions satisfying this condition is provided, yielding several infinite classes of binary SO singly-even minimal linear codes with few weights. Finally, we would like to emphasize that using the methods in this paper, we can construct more binary linear codes that are SO, singly-even, minimal, violating the AB condition, and with few weights at the same time.

Problem

• Characterize binary self-orthogonal singly-even linear codes
• Construct infinite classes of singly-even SO minimal codes
• Develop conditions for Boolean functions generating SO singly-even codes

Innovation

• Characterizes binary SO singly-even linear codes
• Generates SO codes from two known codes
• Constructs Boolean functions for SO singly-even codes