부산대학교 도서관

In 2012, motivated by applications to node failures in large-scale distributed storage systems, P. Gopalan et al. [IEEE Trans. Inform. Theory {\bf 58} (2012), no.~11, 6925--6934; MR2991819] introduced the concept of codes ``with locality", or locally recoverable codes, meaning codes where there exists an integer $r\geq 1$ such that, for any $i$, the value of the $i$-th entry of a codeword may be recovered from the knowledge of values of the entries in a set formed by at most $r$ other coordinates of the codeword. If for each entry of a codeword there are $t\geq 2$ disjoint such sets then the code is said to be locally recoverable ``with availability $t$". \par In the paper under review the authors present constructions of locally recoverable codes, some of them with availability. The locally recoverable codes are obtained using data from maximal curves introduced by D.~C. Skabelund [see Proc. Amer. Math. Soc. {\bf 146} (2018), no.~2, 525--540; MR3731688], which are cyclic extensions of the Suzuki and Ree curves. For each curve a code is constructed as an evaluation code, where elements in a certain vector space contained in the function field of the curve are evaluated in a certain set of points of the curve, and the code is the image of the evaluation morphism. Locally recoverable codes with availability are then constructed as the product of codes obtained using the previous construction. \par The paper ends with a section containing numerical examples.