Sparse and Balanced MDS Codes Over Small Fields

• Published in: IEEE transactions on information theory Vol. 68; no. 8; pp. 5112 - 5125
• Main Authors: Chen, Tingting, Zhang, Xiande
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Codes; Complexity theory; constrained generator matrices; finite fields; Generators; MDS codes; Polynomials; Reed-Solomon codes; Security; Sparse matrices; Storage systems; Wireless sensor networks
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2022.3162524

Maximum Distance Separable (MDS) codes with a sparse and balanced generator matrix are appealing in distributed storage systems for balancing and minimizing the computational load. Such codes have been constructed via Reed-Solomon codes over large fields. In this paper, we focus on small fields. We prove that there exists an <inline-formula> <tex-math notation="LaTeX">[n,k]_{q} </tex-math></inline-formula> MDS code that has a sparse and balanced generator matrix for any <inline-formula> <tex-math notation="LaTeX">q\geq n-1 </tex-math></inline-formula> provided that <inline-formula> <tex-math notation="LaTeX">n\leq 2k </tex-math></inline-formula>, by designing several algorithms with complexity running in polynomial time in <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>. For the case <inline-formula> <tex-math notation="LaTeX">n>2k </tex-math></inline-formula>, we give some constructions for <inline-formula> <tex-math notation="LaTeX">q=n=p^{s} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">k=p^{e}m </tex-math></inline-formula> based on sumsets, when <inline-formula> <tex-math notation="LaTeX">e\leq s-2 </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">m\leq p-1 </tex-math></inline-formula>, or <inline-formula> <tex-math notation="LaTeX">e=s-1 </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">m < \frac {p}{2} </tex-math></inline-formula>.