Shortest self-orthogonal embeddings of binary linear codes

2025-11-07

AI Summary

This paper investigates the shortest self-orthogonal embedding length problem for binary linear codes—i.e., the minimum number of columns to append to a given generator matrix so that the extended code becomes self-orthogonal (or self-dual). We propose a novel method combining structural analysis of the code’s kernel with systematic generator matrix extension. This approach enables, for the first time, direct construction of self-dual codes from Hamming codes. We design two dedicated algorithms for Hamming and Reed–Muller codes, yielding new self-dual codes including [22,11,6] and [52,26,8]. Furthermore, we construct four optimal self-orthogonal codes with previously unattained parameters: [91,8,42], [98,8,46], [114,8,54], and [191,8,94], substantially expanding the known table of optimal self-orthogonal codes.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Machine Learning: Unsupervised & Self-Supervised Learning
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $mathcal C$, or equivalently, the minimum number of columns that must be added to a generator matrix of $mathcal C$ to form a generator matrix of a self-orthogonal code. In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. We focus on the examples of Hamming codes and Reed-Muller codes. We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes $mathcal H_r$. Using these algorithms, we construct a self-dual $[22, 11, 6]$ code, called the shortened Golay code, from the binary $[15, 11, 3]$ Hamming code $mathcal H_4$, and construct a self-dual $[52, 26, 8]$ code from the binary $[31, 26, 3]$ Hamming code $mathcal H_5$. We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91, 8, 42],, [98, 8, 46],,[114, 8, 54]$, and $[191, 8, 94]$.

Problem

• Determining shortest self-orthogonal embeddings of binary linear codes
• Using code hull properties to find minimum embedding lengths
• Constructing optimal self-orthogonal codes from Hamming and Reed-Muller codes

Innovation

• Using hull properties to determine shortest self-orthogonal embeddings
• Proposing two algorithms to construct self-dual codes from Hamming codes
• Applying shortest embeddings to generate optimal self-orthogonal codes