Shortest LCD embeddings of binary, ternary and quaternary linear codes

2026-01-28

AI Summary

This study addresses the problem of embedding an arbitrary linear code into an LCD (Linear Complementary Dual) code with optimal parameters in the shortest possible way. By analyzing the minimum number of columns that must be appended to a generator matrix, the work provides the first complete characterization of the minimal-length embedding structure from linear codes into LCD codes. Leveraging tools from algebraic coding theory and matrix extension techniques, the authors systematically construct a series of LCD codes with high minimum distances. As concrete applications, new optimal LCD codes are obtained for binary, ternary, and quaternary Hamming codes, including ternary $[23,4,14]$, $[23,5,12]$, $[24,6,12]$, $[25,5,14]$, and quaternary $[21,10,8]$ codes, each achieving a minimum distance one unit greater than previously known results.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

In the recent years, there has been active research on self-orthogonal embeddings of linear codes since they yielded some optimal self-orthogonal codes. LCD codes have a trivial hull so they are counterparts of self-orthogonal codes. So it is a natural question whether one can embed linear codes into optimal LCD codes. To answer it, we first determine the number of columns to be added to a generator matrix of a linear code in order to embed the given code into an LCD code. Then we characterize all possible forms of shortest LCD embeddings of a linear code. As examples, we start from binary and ternary Hamming codes of small lengths and obtain optimal LCD codes with minimum distance 4. Furthermore, we find new ternary LCD codes with parameters including $[23, 4, 14]$, $[23, 5, 12]$, $[24, 6, 12]$, and $[25, 5, 14]$ and a new quaternary LCD $[21, 10, 8]$ code, each of which has minimum distance one greater than those of known codes. This shows that our shortest LCD embedding method is useful in finding optimal LCD codes over various fields.

Problem

• LCD codes
• linear codes
• shortest embeddings
• optimal codes
• minimum distance

Innovation

• LCD codes
• shortest embedding
• linear codes
• optimal codes
• minimum distance