New Constant Dimension Codes From the Inserting Mixed Dimension Construction and Multilevel Construction

TL;DR

This paper introduces new methods for constructing constant dimension codes using inserting and multilevel techniques, leading to improved lower bounds for code sizes in network error correction.

Contribution

It provides new inserting constructions and enhancements to existing methods by combining matrix blocks and rank-metric codes, resulting in better CDC bounds.

Findings

New lower bounds for CDCs surpass previous best-known results.

Effective combining of small CDCs and rank-metric codes in constructions.

Improved multilevel construction based on Ferrers diagram rank-metric codes.

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.