Embedding linear codes over Z4 into self-orthogonal codes

2026-06-07

AI Summary

This study addresses the problem of determining the minimal length required to embed linear codes over ℤ₄ into self-orthogonal codes. By integrating algebraic coding theory, combinatorial bound analysis, and algorithmic constructions, the work establishes tight lower bounds and introduces a systematic classification framework. The main contributions include the first complete determination of the minimal doubly-even self-orthogonal embedding length for binary linear codes, an explicit algorithmic construction for all shortest self-orthogonal embeddings of free ℤ₄-linear codes under specific conditions, the exact computation of the minimal self-orthogonal embedding length for the quaternary Preparata code, and the discovery of twelve new ℤ₄-linear codes whose Lee distances surpass the best-known entries in the Aydin database.

Abstract

The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.

Problem

• self-orthogonal embedding
• linear codes over Z4
• shortest embedding length
• doubly even codes
• Lee distance

Innovation

• self-orthogonal embedding
• linear codes over Z4
• doubly even codes
• Lee distance
• Preparata codes