Construction of MRD Codes Based on Circular-Shift Operations

Most well-known constructions of (N×n, qNk, d) maximum rank distance (MRD) codes rely on the arithmetic of FqN, whose increasing complexity with largerNhinders parameter selection and practical implementation. In this work, based on circular-shift operations, we present a construction of (J×n, qJk, d) MRD codes with efficient encoding, whereJequals to the Euler’s totient function of a definedLsubject to gcd(q,L) = 1. The proposed construction is performed entirely over Fqand avoids the arithmetic of FqJ. Furthermore, we characterize the constructed MRD codes, Gabidulin codes and twisted Gabidulin codes using a set ofq-linearized polynomials over the row vector space FNq, and clarify the inherent difference and connection among these code families. Specifically, for the caseJ≠mL, wheremLdenotes the multiplicative order ofqmoduloL, 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 caseJ=mL, we prove that every constructed (J×n, qJk, d) MRD code coincides with a (J×n, qJk, d) Gabidulin code, so that an equivalent circular-shift- based construction of Gabidulin codes is obtained which operates directly over Fqand avoids the arithmetic of FqJ. 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 (mL×n, qmLk, d) Gabidulin codes. Whenq= 2,Lis prime andn≤mL, it is analyzed that generating a codeword of the proposed ((L−1)×n, 2(L−1)k,d) MRD codes requiresO(nkL)exclusive OR (XOR) operations, while generating a codeword of ((L− 1) ×n, 2(L−1)k,d) Gabidulin codes, based on customary construction, requiresO(nkL2)XOR operations.