Bounds on Binary Niederreiter-Rosenbloom-Tsfasman 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.