Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq
Information Theory
Summary
The authors study the shortest ways to embed certain types of linear codes called self-orthogonal and LCD codes over a special ring combining two copies of a finite field. They use a technique to break down complex matrices into simpler ones over the finite field, reducing the problem to classifying these simpler matrices. The paper provides exact formulas for the shortest self-orthogonal embeddings and shows that all such codes with a certain property come from these shortest embeddings. They also fully describe the shortest LCD embeddings and give examples where these embeddings lead to very good related codes over the finite field.
Authors
Junmin An, Jon-Lark Kim
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 $\mathbb{F}_q$.