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This paper focuses on constructions of MDS self-dual codes from (extended) generalized Reed-Solomon (GRS) codes. Let $q = r^{2}$ be an odd prime power. We show that, there exists a $q$ -ary self-dual (extended) GRS code for each even length in the range $[{2r,3r-3}]$ , and for each singly even length in the range $[3r-1,4r]$. This extends the only known consecutive range $[2,2r]$ to $[{2,3r-3}]$ for this case. Furthermore, our general constructions provide many MDS self-dual codes with new parameters which, to the best of our knowledge, were not reported before. [ABSTRACT FROM AUTHOR]

As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,\delta)$ -locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. are considered. Through parity-check matrix approach, constructions of both optimal $(r,\delta)$ -LRCs with all symbol locality ($(r,\delta)_{a}$ -LRCs) and optimal $(r,\delta)$ -LRCs with information locality ($(r,\delta)_{i}$ -LRCs) are provided. As a generalization of a work of Xing and Yuan, these constructions are built on a connection between sparse hypergraphs and optimal $(r,\delta)$ -LRCs. With the help of constructions of large sparse hypergraphs, the lengths of codes obtained from our construction can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least $3\delta +1$. As two applications, optimal H-LRCs with super-linear lengths and GSD codes with unbounded lengths are also constructed. [ABSTRACT FROM AUTHOR]