Generalized Reed-Solomon Codes with Sparsest and Balanced Generator Matrices

We prove that for any positive integers $n$ and $k$ such that $n\!\geq\! k\!\geq\! 1$, there exists an $[n,k]$ generalized Reed-Solomon (GRS) code that has a sparsest and balanced generator matrix (SBGM) over any finite field of size $q\!\geq\! n\!+\!\lceil\frac{k(k-1)}{n}\rceil$, where sparsest means that each row of the generator matrix has the least possible number of nonzeros, while balanced means that the number of nonzeros in any two columns differ by at most one. Previous work by Dau et al (ISIT'13) showed that there always exists an MDS code that has an SBGM over any finite field of size $q\geq {n-1\choose k-1}$, and Halbawi et al (ISIT'16, ITW'16) showed that there exists a cyclic Reed-Solomon code (i.e., $n=q-1$) with an SBGM for any prime power $q$. Hence, this work extends both of the previous results

Balanced Sparsest Generator Matrices for MDS Codes

We show that given $n$ and $k$, for $q$ sufficiently large, there always exists an $[n, k]_q$ MDS code that has a generator matrix $G$ satisfying the following two conditions: (C1) Sparsest: each row of $G$ has Hamming weight $n - k + 1$; (C2) Balanced: Hamming weights of the columns of $G$ differ from each other by at most one.Comment: 5 page

Optimal Locally Repairable Linear Codes

Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n, k, d] code having all-symbol (r, \delta})-locality, denoted as (r, {\delta})a, is considered optimal if it also meets the minimum Hamming distance bound. The existing results on the existence and the construction of optimal (r, {\delta})a codes are limited to only the special case of {\delta} = 2, and to only two small regions within this special case, namely, m = 0 or m >= (v+{\delta}-1) > ({\delta}-1), where m = n mod (r+{\delta}-1) and v = k mod r. This paper investigates the existence conditions and presents deterministic constructive algorithms for optimal (r, {\delta})a codes with general r and {\delta}. First, a structure theorem is derived for general optimal (r, {\delta})a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and {\delta} is divided into eight different cases (regions) with regard to the specific relations of these parameters. For two cases, it is rigorously proved that no optimal (r, {\delta})a could exist. For four other cases the optimal (r, {\delta})a codes are shown to exist, deterministic constructions are proposed and the lower bound on the required field size for these algorithms to work is provided. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a considerably smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on (r, {\delta})a codes, leaving only two cases in which the existence of optimal codes are yet to be determined.Comment: Under Revie