Generalized bilateral multilevel construction for constant dimension codes from parallel mixed dimension construction

TL;DR

This paper presents a new generalized bilateral multilevel construction method that enhances parallel mixed dimension constructions, leading to the creation of larger constant dimension codes with improved parameters for network coding applications.

Contribution

It introduces criteria for bilateral identifying vectors and combines them with a generalized multilevel construction to improve existing constant dimension code constructions.

Findings

Constructed new CDCs surpassing previous best-known codes.

Provided criteria for selecting bilateral identifying vectors.

Enhanced the efficiency of parallel mixed dimension constructions.

Abstract

Constant dimension codes (CDCs), as special subspace codes, have received extensive attention due to their applications in random network coding. The basic problem of CDCs is to determine the maximal possible size $A_q(n,d,\{k\})$ for given parameters $q, n, d$, and $k$. This paper introduces criteria for choosing appropriate bilateral identifying vectors compatible with the parallel mixed dimension construction (Des. Codes Cryptogr. 93(1):227--241, 2025). We then utilize the generalized bilateral multilevel construction (Des. Codes Cryptogr. 93(1):197--225, 2025) to improve the parallel mixed dimension construction efficiently. Many new CDCs that are better than the previously best-known codes are constructed.