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

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 ${\mathbb{F}}_q$.