Type-preserving shortest embeddings over finite chain rings

Investigate shortest self-orthogonal and LCD embedding problems over rings under the additional requirement that the type of the original linear code over \(\mathbb{F}_q+u\mathbb{F}_q\) be preserved, thereby determining the minimum embedding lengths and characterizing the corresponding embeddings.

Background

The paper determines shortest self-orthogonal and LCD embeddings of linear codes over $R=\mathbb{F}_q+u\mathbb{F}_q$ without requiring the embedding to preserve the type $\{k_1,k_2\}$ of the original code. Since appending columns can alter this type, the conclusion identifies type preservation as a concrete unresolved direction for future work. The open problem is to develop shortest-embedding results subject to this additional structural constraint.