Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of MRD Codes Based on Circular-Shift Operations

Zhe Zhai, Sheng Jin, Qifu Tyler Sun, Zongpeng Li

TL;DR

It is proved that under some parameter settings, the constructed MRD codes are equivalent to a generalization of Gabidulin codes obtained by summing and concatenating several $(m_L \times n, q^{m_Lk}, d)$ Gabidulin codes.

Abstract

Most well-known constructions of $(N \times n, q^{Nk}, d)$ maximum rank distance (MRD) codes rely on the arithmetic of $\mathbb{F}_{q^N}$, whose increasing complexity with larger $N$ hinders parameter selection and practical implementation. In this work, based on circular-shift operations, we present a construction of $(J \times n, q^{Jk}, d)$ MRD codes with efficient encoding, where $J$ equals to the Euler's totient function of a defined $L$ subject to $\gcd(q, L) = 1$. The proposed construction is performed entirely over $\mathbb{F}_q$ and avoids the arithmetic of $\mathbb{F}_{q^J}$. We further characterize the constructed MRD codes, Gabidulin codes and twisted Gabidulin codes using a set of $q$-linearized polynomials over the row vector space $\mathbb{F}_{q}^N$, and clarify their inherent difference and connection. For the case $J \neq m_L$, where $m_L$ denotes the multiplicative order of $q$ modulo $L$, we show that the proposed MRD codes, in a family of settings, are different from any Gabidulin code and any twisted Gabidulin code. For the case $J = m_L$, we prove that every constructed $(J \times n, q^{Jk}, d)$ MRD code coincides with a $(J \times n, q^{Jk}, d)$ Gabidulin code, yielding an equivalent circular-shift-based construction that operates directly over $\mathbb{F}_q$. In addition, we prove that under some parameter settings, the constructed MRD codes are equivalent to a generalization of Gabidulin codes obtained by summing and concatenating several $(m_L \times n, q^{m_Lk}, d)$ Gabidulin codes. When $q=2$, $L$ is prime and $n\leq m_L$, it is analyzed that generating a codeword of the proposed $((L-1) \times n, 2^{(L-1)k}, d)$ MRD codes requires $O(nkL)$ exclusive OR (XOR) operations, while generating a codeword of $((L-1) \times n, 2^{(L-1)k}, d)$ Gabidulin codes, based on customary construction, requires $O(nkL^2)$ XOR operations.

Construction of MRD Codes Based on Circular-Shift Operations

TL;DR

It is proved that under some parameter settings, the constructed MRD codes are equivalent to a generalization of Gabidulin codes obtained by summing and concatenating several Gabidulin codes.

Abstract

Most well-known constructions of maximum rank distance (MRD) codes rely on the arithmetic of , whose increasing complexity with larger hinders parameter selection and practical implementation. In this work, based on circular-shift operations, we present a construction of MRD codes with efficient encoding, where equals to the Euler's totient function of a defined subject to . The proposed construction is performed entirely over and avoids the arithmetic of . We further characterize the constructed MRD codes, Gabidulin codes and twisted Gabidulin codes using a set of -linearized polynomials over the row vector space , and clarify their inherent difference and connection. For the case , where denotes the multiplicative order of modulo , we show that the proposed MRD codes, in a family of settings, are different from any Gabidulin code and any twisted Gabidulin code. For the case , we prove that every constructed MRD code coincides with a Gabidulin code, yielding an equivalent circular-shift-based construction that operates directly over . In addition, we prove that under some parameter settings, the constructed MRD codes are equivalent to a generalization of Gabidulin codes obtained by summing and concatenating several Gabidulin codes. When , is prime and , it is analyzed that generating a codeword of the proposed MRD codes requires exclusive OR (XOR) operations, while generating a codeword of Gabidulin codes, based on customary construction, requires XOR operations.
Paper Structure (24 sections, 13 theorems, 160 equations, 1 table)

This paper contains 24 sections, 13 theorems, 160 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vector space representation of extension field
  4. $q$-linearized polynomial
  5. Gabidulin codes
  6. Circular-shift-based MRD codes
  7. New construction of rank-metric codes
  8. Explicit construction of $\mathbf{P}$ and $\mathbf{Q}$ for circular-shift-based MRD codes
  9. An alternative construction of $\mathbf{P}$ and $\mathbf{Q}$
  10. Inherent relation and difference between the new codes and Gabidulin codes
  11. An alternative characterization of Gabidulin codes
  12. Relation and difference between the new codes and Gabidulin codes
  13. Generalization of Gabidulin codes
  14. Computational complexity analysis
  15. Concluding Remarks
  16. ...and 9 more sections

Key Result

Lemma 2

Given a $q$-linearized polynomial $L(x)$ over $\mathbb{F}_{q^N}$ and $n$ elements $\beta_0, \ldots, \beta_{n-1} \in \mathbb{F}_{q^N}$, if for some $c_0, \ldots, c_{n-1} \in \mathbb{F}_q$, then $\sum\nolimits_{0 \leq i \leq n-1} c_i\beta_i$ is a root of $L(x)$ and for all $j \geq 0$,

Theorems & Definitions (24)

  • Definition 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Proposition 5
  • Example 1
  • Definition 6
  • Theorem 7
  • Remark
  • Example 2
  • ...and 14 more