🤖 AI Summary
This paper addresses the low lower bound on minimum distance in constant-dimension codes (CDCs). To tackle this, we propose a multi-level construction method based on one-factorizations of complete graphs. Our key innovation lies in designing novel skeleton codes via binary vector transformations induced by one-factorizations, and integrating quasi-pendant block structures with optimal Ferrers diagram rank-metric codes to construct CDCs achieving high minimum distance. The approach systematically refines the multi-level construction framework: while fixing the dimension $k = 6$ and minimum distance $d = 8$, it significantly improves the cardinality lower bounds for code lengths $n in [16,19]$, thereby advancing the state-of-the-art lower bounds on $overline{A}_q(n,8,6)$. This work provides both new theoretical insights and practical construction tools for network coding and random linear network coding over finite fields.
📝 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 $16leq nleq 19.$