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.

💡 Research Summary

The paper addresses a fundamental practical limitation of most maximum rank distance (MRD) code constructions: they rely on arithmetic in the extension field 𝔽_{q^N}, which becomes increasingly cumbersome as N grows, restricting parameter choices and hampering efficient implementation. Inspired by circular‑shift linear network coding, the authors propose a completely field‑grounded construction that works solely over the base field 𝔽_q, eliminating any need for 𝔽_{q^J} arithmetic.

Key parameters are an integer L with gcd(q, L)=1, its Euler totient J = φ(L), and the multiplicative order m_L of q mod L. The construction is valid for any n ≤ m_L (and thus n ≤ J). The code family is defined as

C = { Δ( m·(I_k ⊗ P)·Ψ_{k×n}·(I_n ⊗ Q) ) | m ∈ 𝔽_q^{Jk} },

where

* P ∈ 𝔽_q^{J×L} and Q ∈ 𝔽_q^{L×J} are matrices built by two general methods,
* Ψ_{k×n} is a k × n block matrix whose blocks are L × L circulant matrices over 𝔽_q,
* ⊗ denotes the Kronecker product, and
* Δ maps a J·n dimensional row vector over 𝔽_q to a J × n matrix over 𝔽_q.

Because each block of Ψ is a circulant matrix, multiplication by it reduces to a cyclic shift plus a scalar multiplication, which can be implemented with simple XOR and shift operations. Consequently, encoding a message vector m requires only O(nkL) elementary operations, a dramatic improvement over the O(nkL²) complexity of the classic Gabidulin construction when q = 2 and L is prime.

To place the new codes in the broader MRD landscape, the authors develop a unified polynomial‑based framework. They show that both Gabidulin and twisted Gabidulin codes can be described by a set of q‑linearized polynomials evaluated over the row‑vector space 𝔽_q^J, rather than over the extension field. Within this framework, the circular‑shift construction corresponds to a different evaluation map. The paper proves:

• When J ≠ m_L (i.e., J is not the multiplicative order of q modulo L), there exist families of parameters for which the new codes are not equivalent to any Gabidulin or twisted Gabidulin code (Theorem 12, Proposition 13).
• When J = m_L, every code produced by the construction coincides with a Gabidulin code (Theorem 12). Hence the construction provides an alternative, purely 𝔽_q‑based way to generate Gabidulin MRD codes.
• For certain choices of P and Q, the constructed codes are equivalent to a generalized Gabidulin code obtained by summing and concatenating several (m_L × n, q^{m_Lk}, d) Gabidulin codes (Theorem 16). This generalization has not been previously explored.

The paper also revisits the classical vector‑space representation of 𝔽_{q^N} over 𝔽_q, the definition of q‑linearized polynomials, and the two equivalent formulations of Gabidulin codes (matrix‑based and vector‑based). It highlights the practical difficulty of selecting a basis that enables efficient conversion between a row vector over 𝔽_{q^N} and its matrix representation over 𝔽_q, a problem that the circular‑shift approach sidesteps entirely.

Complexity analysis focuses on the binary case (q = 2) with prime L and n ≤ m_L. The authors demonstrate that generating a codeword of the proposed ((L‑1) × n, 2^{(L‑1)k}, d) MRD code needs only O(nkL) XORs, whereas the traditional Gabidulin method requires O(nkL²) XORs because it must evaluate linearized polynomials in 𝔽_{2^{L‑1}} and perform basis transformations.

In conclusion, the work introduces a novel, implementation‑friendly MRD code construction that either reproduces Gabidulin codes without extension‑field arithmetic (when J = m_L) or yields genuinely new MRD codes (when J ≠ m_L). By leveraging circular‑shift operations and a unified polynomial viewpoint, it expands the design space for rank‑metric codes and offers substantial computational savings, making MRD codes more accessible for cryptography, distributed storage, and network coding applications.