Dickson Matrix Based Construction of Linear Maximum Rank Distance Codes

Shu Liu,Yaxin Zhang

doi:10.1109/access.2020.3037153

Abstract

Maximum rank distance (MRD for short) codes lately attract more attention due to their various applications in storage systems, network coding, cryptography and space time coding. Similar to Reed-Solomon codes in classical coding theory, Gabidulin codes are the most prominent family of MRD codes. Due to their poor performance in list decoding or in constructing McEliece-type cryptosystems, the focus moves from Gabidulin codes to other non-Gabidulin codes. A natural following challenge is then to see if we can construct an infinite family of MRD codes that are not equivalent to Gabidulin codes. In this paper, we utilize Dickson matrices to construct an infinite family of Fq-linear MRD codes. Our codes are characterized by each of their codewords corresponding to a linearized polynomial with leading coefficient determined by one of any other coefficients. The family of codes corresponding to the set of linearized polynomials with leading coefficients dependent on the linear terms provides an extension to both Twisted Gabidulin codes and generalized Twisted Gabidulin codes for dimensions 1 and n-1. Lastly, we also provide some analysis on the equivalence between our proposed codes with some known families of MRD codes.

Highlights

R ANK-metric codes were firstly introduced by Delsarte in [1]
We provide some analysis on the equivalence between our proposed codes with some known families of maximum rank distance (MRD) codes
Gabidulin codes can be viewed as an analog of Reed-Solomon codes for rank-metric and they are defined by evaluating degree-restricted linearized polynomials [12]

Summary

INTRODUCTION

R ANK-metric codes were firstly introduced by Delsarte in [1]. This class of codes has recently attracted attention due to their applications to error control in storage system, network coding, cryptography and space time coding [2]–[10]. (see in [11]) Let C ⊆ Fnq ×n be a linear rankmetric code with the minimum rank distance d and dimension k, one has the following Singleton bound k ≤ n − d + 1. Fixqsi + ηf0qh xqsk : fi ∈ Fqn , i=0 is an Fq-linear MRD code of size qnk, which is called a generalized Twisted Gabidulin code. On in this paper, we will use a set of linearized polynomials to represent a rank-metric code. F. THE DELSARTE’S DUAL Firstly, we define the symmetric bilinear form b on linearized polynomials by n−1 n−1 n−1 b fixqi , gixqi := T r figi , i=0 i=0 i=0 where Tr denotes the absolute trace from Fqn to Fq. Definition 9. We focus on studying the relationship between our family F(i,h) and the existing families

COMPARISON WITH KNOWN FAMILIES OF MRD CODES

COMPARISON WITH OTHER EXISTING FAMILIES

COMMENTS ON CONNECTIONS THROUGH

CONCLUSION AND DISCUSSION