Construction of MRD Codes Based on Circular-Shift Operations

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.