Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq

2026-08-28

AI Summary

本文解决了Fq+uFq上线性码最短自正交和LCD嵌入长度问题，通过矩阵分解及Witt理论方法获得其完整公式。

Abstract

This paper determines the exact lengths of shortest self-orthogonal and LCD embeddings of linear codes over $\mathbb{F}_q+u\mathbb{F}_q$. By decomposing Gram matrices over $\mathbb{F}_q+u\mathbb{F}_q$ into pairs of symmetric matrices over the finite field $\mathbb{F}_q$, the embedding problems are reduced to the congruence classification of symmetric and alternate matrices over finite fields. Complete formulas for the shortest self-orthogonal embedding length are obtained, with two distinct cases arising in both even and odd characteristic. We also show that every self-orthogonal code over $\mathbb{F}_q+u\mathbb{F}_q$ with nonzero free rank can be viewed as a shortest self-orthogonal embedding of another code. We use Witt theory to construct all shortest self-orthogonal embeddings. A complete characterization of shortest LCD embeddings is also established in terms of invertible and arbitrary matrices of prescribed sizes appended to a generator matrix. Examples of self-orthogonal and LCD embeddings with the largest minimum distance for the code considered are also presented, some of whose Gray images are optimal codes over $\mathbb{F}_q$.

Problem

• self-orthogonal
• LCD embeddings
• linear codes
• Fq+uFq
• Gram matrices

Innovation

• self-orthogonal embedding
• LCD embedding
• Gram matrix decomposition
• finite field
• Witt theory