Bounds on Binary Niederreiter-Rosenbloom-Tsfasman LCD codes

TL;DR

This paper investigates bounds and constructions of binary Niederreiter-Rosenbloom-Tsfasman LCD codes, focusing on their maximum minimum distances and providing theoretical bounds and construction methods.

Contribution

It establishes existence and non-existence results for binary NRT-LCD codes, introduces a linear programming bound, and proposes two construction methods.

Findings

Proved existence of binary MDS NRT-LCD codes in certain cases.

Developed a linear programming bound for binary NRT-LCD codes in specific spaces.

Presented two methods for constructing binary NRT-LCD codes.

Abstract

Linear complementary dual codes (LCD codes) are codes whose intersections with their dual codes are trivial. These codes were introduced by Massey in 1992. LCD codes have wide applications in data storage, communication systems, and cryptography. Niederreiter-Rosenbloom-Tsfasman LCD codes (NRT-LCD codes) were introduced by Heqian, Guangku and Wei as a generalization of LCD codes for the NRT metric space $M_{n,s}(\mathbb{F}_{q})$. In this paper, we study LCD$[n\times s,k]$, the maximum minimum NRT distance among all binary $[n\times s,k]$ NRT-LCD codes. We prove the existence (non-existence) of binary maximum distance separable NRT-LCD codes in $M_{1,s}(\mathbb{F}_{2})$. We present a linear programming bound for binary NRT-LCD codes in $M_{n,2}(\mathbb{F}_{2})$. We also give two methods to construct binary NRT-LCD codes.