Multilevel constructions of constant dimension codes based on one-factorization of complete graphs

TL;DR

This paper introduces a new method for constructing constant dimension codes using one-factorization of complete graphs, improving bounds for certain parameters in network coding.

Contribution

It presents a novel multilevel construction approach based on graph factorizations and Ferrers diagram codes, enhancing existing bounds for CDCs.

Findings

Improved lower bounds for A_q(n,8,6) for 16  n  19

Constructed CDCs using skeleton codes derived from one-factorizations

Utilized Ferrers diagram rank metric codes for dimension calculations

Abstract

Constant dimension codes (CDCs) have become an important object in coding theory due to their application in random network coding. The multilevel construction is one of the most effective ways to construct constant dimension codes. The paper is devoted to constructing CDCs by the multilevel construction. Precisely, we first choose an appropriate skeleton code based on the transformations of binary vectors related to the one-factorization of complete graphs; then we construct CDCs by using the chosen skeleton code, where quasi-pending blocks are used; finally, we calculate the dimensions by use of known constructions of optimal Ferrers diagram rank metric codes. As applications, we improve the lower bounds of $\overline{A}_q(n,8,6)$ for $16\leq n\leq 19.$