Optimal LCD codes in dimensions nine and ten

Construct optimal binary LCD codes of dimensions 9 and 10 that attain the corresponding upper bounds for the minimum distance over the parameter ranges investigated in the paper.

Background

The paper applies greedy search to shortest LCD embeddings to construct binary LCD codes with large minimum distance. The search produces codes attaining the derived upper bounds for several parameters with dimensions 6, 7, and 8, including 14 previously unknown optimal codes.

For dimensions 9 and 10, the reported computational results do not include any code attaining the corresponding upper bounds. The authors therefore explicitly leave the construction of such optimal LCD codes as unresolved future work. The problem concerns finding constructions, rather than merely improving the heuristic search results, that reach the stated upper bounds for dimensions 9 and 10.