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.

翻译：大多数已知的$(N \times n, q^{Nk}, d)$最大秩距离（MRD）码的构造依赖于$\mathbb{F}_{q^N}$的算术运算，其复杂度随$N$增大而增加，这阻碍了参数选择和实际实现。本文基于循环移位操作，提出了一种具有高效编码的$(J \times n, q^{Jk}, d)$ MRD码构造方法，其中$J$等于一个满足$\gcd(q, L) = 1$的给定$L$的欧拉函数值。所提出的构造完全在$\mathbb{F}_q$上进行，避免了$\mathbb{F}_{q^J}$的算术运算。我们进一步利用行向量空间$\mathbb{F}_{q}^N$上的一组$q$-线性化多项式来刻画所构造的MRD码、Gabidulin码和扭曲Gabidulin码，并阐明了它们的内在差异与联系。对于$J \neq m_L$的情况（其中$m_L$表示$q$模$L$的乘法阶），我们证明了所提出的MRD码在一系列参数设置下不同于任何Gabidulin码和任何扭曲Gabidulin码。对于$J = m_L$的情况，我们证明了每个构造的$(J \times n, q^{Jk}, d)$ MRD码都与一个$(J \times n, q^{Jk}, d)$ Gabidulin码重合，从而得到了一种直接在$\mathbb{F}_q$上操作的、基于循环移位的等价构造。此外，我们证明在某些参数设置下，所构造的MRD码等价于通过对多个$(m_L \times n, q^{m_Lk}, d)$ Gabidulin码进行求和与拼接而得到的Gabidulin码的推广形式。当$q=2$、$L$为素数且$n\leq m_L$时，分析表明生成所提出的$((L-1) \times n, 2^{(L-1)k}, d)$ MRD码的一个码字需要$O(nkL)$次异或（XOR）运算，而基于常规构造生成$((L-1) \times n, 2^{(L-1)k}, d)$ Gabidulin码的一个码字则需要$O(nkL^2)$次异或运算。