Application of optimal p-ary linear codes to alphabet-optimal locally repairable codes

Abstract

Linear codes have widespread applications in data storage systems. There are two major contributions in this paper. We first propose infinite families of optimal or distance-optimal linear codes over Fp constructed from projective spaces. Moreover, a necessary and sufficient condition for such linear codes to be Griesmer codes is presented. Secondly, as an application in data storage systems, we investigate the locality of the linear codes constructed. Furthermore, we show that these linear codes are alphabet-optimal locally repairable codes with locality 2.

Antimicrobial Resistance Analysis of Escherichia coli Isolated from Pigeons in Qingdao, Shandong Province, China

“…From the finite geometry point of view, we give a simple but useful sufficient condition to guarantee a linear code to be a (2, δ)-LRC (see Theorem 5). We generalize some results proposed by Luo et al [18] (see Theorems 6, 7 and 8) and Silberstein et al [11] (see Theorem 10). In particular, we extend the p-ary linear codes presented in [18] to the qary linear codes, where p is a prime and q is the power of p, and determine their locality in some cases.…”

“…We generalize some results proposed by Luo et al [18] (see Theorems 6, 7 and 8) and Silberstein et al [11] (see Theorem 10). In particular, we extend the p-ary linear codes presented in [18] to the qary linear codes, where p is a prime and q is the power of p, and determine their locality in some cases. Motivated by Silberstein's work on r-LRCs, we utilize Krawtchouk polynomials to determine the parameters of some punctured simplex codes.…”

“…Hyun et al [14] constructed infinite families of binary Griesmer codes punctured by unions of projective spaces, and Luo et al [18] obtained similar results of linear codes over F p . In the following, we extend their results to general q-ary codes.…”

“…In particular, to ensure the code C D c to be a 2-LRC, it only needs to satisfy that |D| ≤ q m−1 − 2. Thus our method is simpler than that in [18].…”

New Constructions of Optimal Cyclic (r, δ) Locally Repairable Codes From Their Zeros

“…More recently, Hyun et al [17] constructed infinite families of binary Griesmer codes by utilizing unions of projective subspaces. This construction was later generalized to the p-ary case by Luo et al [24].…”