Abstract-Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n, k, d] code having all-symbol (r, δ)-locality, denoted as (r, δ)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, δ)a codes are limited to only the special case of δ = 2, and to only two small regions within this special case, namely, m = 0 or m ≥ (v +δ −1) > (δ −1), where m = n mod (r+δ −1) and v = k mod r. This paper investigates the existence conditions and presents deterministic constructive algorithms for optimal (r, δ)a codes with general r and δ. First, a structure theorem is derived for general optimal (r, δ)a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and δ 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, δ)a could exist. For four other cases the optimal (r, δ)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, δ)a codes, leaving only two cases in which the existence of optimal codes are yet to be determined.

We study the existence over small fields of Maximum Distance Separable (MDS) codes with generator matrices having specified supports (i.e. having specified locations of zero entries). This problem unifies and simplifies the problems posed in recent works of Yan and Sprintson (NetCod'13) on weakly secure cooperative data exchange, of Halbawi et al. (arxiv'13) on distributed Reed-Solomon codes for simple multiple access networks, and of Dau et al. (ISIT'13) on MDS codes with balanced and sparse generator matrices. We conjecture that there exist such [n, k]q MDS codes as long as q ≥ n + k − 1, if the specified supports of the generator matrices satisfy the so-called MDS condition, which can be verified in polynomial time. We propose a combinatorial approach to tackle the conjecture, and prove that the conjecture holds for a special case when the sets of zero coordinates of rows of the generator matrix share with each other (pairwise) at most one common element. Based on our numerical result, the conjecture is also verified for all k ≤ 7. Our approach is based on a novel generalization of the well-known Hall's marriage theorem, which allows (overlapping) multiple representatives instead of a single representative for each subset. arXiv:1401.3807v1 [cs.IT] 16 Jan 2014 • | ∪ i∈I R i | ≥ n − k + |I|, for all nonempty subsets I ⊆ [k]. • |R i | = n − k + 1, for all i ∈ [k]. Moreover, such subsets R i can be found in polynomial time.

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.

Locally repairable codes (LRC) for distribute storage allow two approaches to locally repair multiple failed nodes: 1) parallel approach, by which each newcomer access a set of r live nodes (r is the repair locality) to download data and recover the lost packet; and 2) sequential approach, by which the newcomers are properly ordered and each newcomer access a set of r other nodes, which can be either a live node or a newcomer ordered before it. An [n, k] linear code with locality r and allows local repair for up to t failed nodes by sequential approach is called an (n, k, r, t)-exact locally repairable code (ELRC).In this paper, we present a family of binary codes which is equivalent to the direct product of m copies of the [r+1, r] singleparity-check code. We prove that such codes are (n, k, r, t)-ELRC with n = (r + 1) m , k = r m and t = 2 m − 1, which implies that they permit local repair for up to 2 m − 1 erasures by sequential approach. Our result shows that the sequential approach has much bigger advantage than parallel approach.

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