New Systematic MDS Array Codes With Two Parities

• Published in: IEEE open journal of the Communications Society Vol. 5; pp. 6329 - 6342
• Main Authors: Ma, Lan, Zhou, Liyang, Liu, Shaoteng, Chen, Xiangyu, Sun, Qifu
• Format: Journal Article
• Language: English
• Published: New York IEEE 2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Array code; Arrays; Codecs; Codes; Coding; Complexity; Complexity theory; Decoding; decoding complexity; Encoding; encoding complexity; Generators; Kernel; Lower bounds; maximum distance separable (MDS) code; Permutations; Sun; Systematics; update complexity
• ISSN: 2644-125X; 2644-125X
• DOI: 10.1109/OJCOMS.2024.3468873

Row-diagonal parity (RDP) code is a classical <inline-formula> <tex-math notation="LaTeX">(k+2,~k) </tex-math></inline-formula> systematic maximum distance separable (MDS) array code with <inline-formula> <tex-math notation="LaTeX">k \leq L-1 </tex-math></inline-formula> under sub-packetization level <inline-formula> <tex-math notation="LaTeX">l = L-1 </tex-math></inline-formula>, where L is a prime integer. When <inline-formula> <tex-math notation="LaTeX">k = L-1 </tex-math></inline-formula>, its encoding requires <inline-formula> <tex-math notation="LaTeX">2-{}\frac {2}{k} </tex-math></inline-formula> XORs per original data bit, which exactly achieves theoretical optimal lower bound. In this paper, we present three new constructions of <inline-formula> <tex-math notation="LaTeX">(k+2,~k) </tex-math></inline-formula> systematic MDS array codes. First, under sub-packetization level <inline-formula> <tex-math notation="LaTeX">l = 4 </tex-math></inline-formula>, we novelly design a <inline-formula> <tex-math notation="LaTeX">(17,~15) </tex-math></inline-formula> array code <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}}_{1} </tex-math></inline-formula>, where k can reach the largest possible value to satisfy the MDS property. Moreover, when <inline-formula> <tex-math notation="LaTeX">k \leq 7 </tex-math></inline-formula>, the encoding complexity of its subcodes can exactly achieve the theoretical optimal <inline-formula> <tex-math notation="LaTeX">2-{}\frac {2}{k} </tex-math></inline-formula> XORs per original data bit, and likewise, the decoding complexity of the subcodes with <inline-formula> <tex-math notation="LaTeX">k \leq 4 </tex-math></inline-formula> is also exactly optimal. Under sub-packetization level <inline-formula> <tex-math notation="LaTeX">l = L-1 </tex-math></inline-formula> with certain primes L, the second construction yields an MDS array code <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}}_{2} </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">k \leq {}\frac {L(L-1)}{2} </tex-math></inline-formula>, and the encoding complexity of <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}}_{2} </tex-math></inline-formula> is also exactly optimal for <inline-formula> <tex-math notation="LaTeX">k = L-1 </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">2L-3 </tex-math></inline-formula>. Furthermore, based on bit permutation, the third MDS array code <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}}_{3} </tex-math></inline-formula> is obtained with <inline-formula> <tex-math notation="LaTeX">k \leq L(L-1) </tex-math></inline-formula> under sub-packetization level <inline-formula> <tex-math notation="LaTeX">l = 2(L-1) </tex-math></inline-formula> with certain primes L. In particular, as an extension of <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}}_{2} </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}}_{3} </tex-math></inline-formula> exactly achieves the optimal encoding complexity for <inline-formula> <tex-math notation="LaTeX">k = 2(2L-3) </tex-math></inline-formula>, which does not hold for other array codes in the literature.