Matrix representation of the finite field GF(pm) represents elements in GF(pm) by m × m matrices over GF(p) so that the arithmetic of GF(pm) can be interpreted as the arithmetic among matrices over GF(p). It is a commonly used method to transform a (scalar) linear operation over GF(pm) into a vector linear operation over GF(p)m, and has been practically adopted by several coding libraries to implement maximum distance separable (MDS) codes over GF(pm). Previous construction of matrix representation of GF(pm) stems from the companion matrix of an irreducible polynomial over GF(p). In this paper, we introduce a new approach to construct matrix representation denoted by $\mathcal{B}$ based on the cyclic permutation matrix C, and prove the isomorphism between $\mathcal{B}$ and GF(pm). As every matrix in B can be expressed in the form of Gf (C)H with carefully designed matrices G and H over GF(p) and a polynomial f(x) over GF(p), we further show that the maximum number of nonzero terms in f(x) can be potentially reduced from m. By utilizing this property, we demonstrate that the computational complexity of the linear coding process based on matrix representation $\mathcal{B}$ can be reduced compared with the one based on standard matrix representation, and the reduction rate is up to 13.47% among the instances illustrated in this paper.

Most well-known constructions of (N,n,d) maximum rank distance (MRD) codes rely on the arithmetic of ${\mathbb{F}_{{q^N}}}$, whose increasing computational complexity with larger N hinders parameter selection and practical implementation. In this work, based on circular-shift operations, we present an efficient construction of (J,n,d) MRD codes over ${\mathbb{F}_q}$ with n ≤ mL, where q is prime, L is a positive integer satisfying gcd(q,L) = 1, mLdenotes the multiplicative order of q modulo L and J equals to the Euler’s totient function of L. The proposed construction is performed entirely over ${\mathbb{F}_q}$ and avoids the arithmetic of ${\mathbb{F}_{{q^J}}}$. 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,d) Gabidulin codes. In this sense, the constructed MRD codes differ from conventional Gabidulin codes. In the special case J = mL, we prove that every (mL,n,d) circular-shift-based MRD code coincides with an (mL,n,d) Gabidulin code. Last, when q = 2, L is prime and n ≤ mL, it is analyzed that generating a codeword of the proposed (L −1, n,d) MRD codes requires O(nkL) XOR operations, while generating a codeword of (L −1, n,d) Gabidulin codes, based on customary construction, requires O(nkL2) XOR operations.

Row-Diagonal-Parity (RDP) codes and EVENODD codes are classical systematic array codes and most attention in the literature has been on the generalization of RDP codes. In this work, as generalization of not only RDP codes but also EVENODD codes, we present new construction of$\phi (L)$-dimensional$(k+r, k)$systematic array codes with$r \leq 4$, where L is an odd integer and$\phi (L)$represents the Euler’s totient function of L. We explicitly characterize sufficient conditions on the selection of L to make the codes maximum distance separable (MDS). Compared with EVENODD codes and RDP codes, the largest k that can be supported by the new codes is nearly doubled, and the asymptotic encoding complexity of the new codes is same, that is, asymptotically approaches r XORs per original data bit with increasing L and k. Moreover, for prime L,$r = 2$and$k = 2L-3$, the new code exactly achieves the optimal encoding complexity. For the case$r = 4$, the largest k that can be supported by the new codes is larger than the recently proposed so-called Variants of Extended Shortened Independent-Parity (V-ESIP) systematic array code in a number of code dimension selections, and meanwhile, the obtained explicit conditions on L to guarantee the MDS property of the new codes also apply to classical EVENODD codes and RDP codes, but are more general than well known explicit ones in the literature. The decoding process of the new array codes is also discussed. In particular, the$r\times r$block inverse matrix involved in decoding is explicitly characterized, which applies to all MDS array codes generalized from RDP or EVENODD codes in the literature.

Given a prime L, we present a new construction of (L−1)-dimensional (k+r,k) systematic array codes with r ≤ 4, and concretely characterize sufficient conditions on the selection of L to guarantee the codes’ MDS property. The largest possible k that can be supported by the new MDS array codes is 2L−4, nearly twice as large as that supported by classical MDS array codes such as EVENODD codes and RDP codes. Moreover, the number of XORs per original data bit required in encoding of the new codes asymptotically approaches r with increasing k and L, same as EVENODD codes and RDP codes. In addition, for the case r = 4, the explicit conditions on L we obtain to guarantee the new codes’ MDS property can also be used to guarantee the MDS property of EVENODD codes and RDP codes, but are more general than the well known ones in the literature.