New Constant Dimension Codes From the Inserting Mixed Dimension Construction and Multilevel Construction
Han Li, Fang-Wei Fu
TL;DR
The paper tackles the problem of determining the maximal size $A_q(n,d,\{k\})$ of constant-dimension codes by enhancing the mixed-dimension construction MixDD with inserting constructions and multilevel lifting. It provides a sufficient condition to augment MDC-based CDCs, introduces three inserting constructions, and fuses these with rank-restricted Ferrers diagram codes to significantly enlarge code sizes. Through extensive use of RFDRMCs and FDRMCs within the multilevel framework, the authors derive numerous new lower bounds—at least 63 in total—surpassing previously best-known results for a wide range of parameters. The work demonstrates how combining MDCs, RRMCs, and multilevel lifting yields practical CDCs with improved performance for random network coding, while outlining future directions in RFDRMC and MDDC theory to push further gains.
Abstract
Constant dimension codes (CDCs) are essential for error correction in random network coding. A fundamental problem of CDCs is to determine their maximal possible size for given parameters. Inserting construction and multilevel construction are two effective techniques for constructing CDCs. We first provide a sufficient condition for a subspace to be added to the code from the mixed dimension construction in Lao et al. (IEEE Trans. Inf. Theory 69(7): 4333-4344, 2023). By appropriately combining matrix blocks from small CDCs and rank-metric codes, we introduce three inserting constructions based on the mixed dimension construction. Furthermore, the mixed dimension construction and these inserting constructions are improved by the multilevel construction that is based on lifting rank-restricted Ferrers diagram rank-metric codes. Our constructions yield some new lower bounds for CDCs, which are superior to the previously best-known ones.