Multilevel constructions of constant dimension codes based on one-factorization of complete graphs
Dengming Xu, Mengmeng LI
TL;DR
This work advances constant-dimension code construction by a multilevel approach that leverages skeleton codes derived from one-factorizations of complete graphs and augments them with quasi-pending blocks and Ferrers diagram rank-metric codes. The authors develop a concrete construction that yields $((n+3)t,4t,3t)_q$ CDCs for even $t$ and $n\ge 5$, under certain field-size conditions, and provide explicit lower bounds for $\overline{A}_{q}((n+3)t,4t,3t)$ with detailed case analyses. Applying these results to $n$ in $\{16,17,18,19\}$, they obtain improved lower bounds for $\overline{A}_{q}(n,8,6)$, including new corollaries and refined bounds that surpass previously known values. The work also articulates explicit calculations of the contributions from skeleton classes and quasi-pending blocks, highlighting both the method's strength and remaining diagrams where optimal FDRMCs are yet to be established, pointing to future refinements in CDC construction.
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 $16\leq n\leq 19.$