Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n, k, d] code having all-symbol (r, )-locality, denoted as (r, )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, )a codes are limited to only the special case of = 2, and to only two small regions within this special case, namely, m = 0 or m >= (v+ -1) > ( -1), where m = n mod (r+ -1) and v = k mod r. This paper investigates the existence conditions and presents deterministic constructive algorithms for optimal (r, )a codes with general r and . First, a structure theorem is derived for general optimal (r, )a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and 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, )a could exist. For four other cases the optimal (r, )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, )a codes, leaving only two cases in which the existence of optimal codes are yet to be determined.
It was proved in @cite_2 that there exists optimal locally repairable linear codes when @math and @math . Under the condition that @math , a construction method of optimal locally repairable vector codes was proposed in @cite_7 , where maximal rank distance (MRD) codes were used along with MDS array codes. For the special case of @math , Tamo @cite_5 proposed an explicit construction of optimal LRCs when @math or @math Except for the special case that @math , no results are known about whether there exists optimal @math code when @math .
@cite_5 @cite_7 @cite_2
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This paper presents a new explicit construction for locally repairable codes (LRCs) for distributed storage systems which possess all-symbols locality and maximal possible minimum distance, or equivalently, can tolerate the maximal number of node failures. This construction, based on maximum rank distance (MRD) Gabidulin codes, provides new optimal vector and scalar LRCs. In addition, the paper also discusses mechanisms by which codes obtained using this construction can be used to construct LRCs with efficient repair of failed nodes by combination of LRC with regenerating codes. An @math maximum distance separable (MDS) array code of length @math , dimension @math , and sub-packetization @math is formed of @math matrices over a finite field @math , with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of @math surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with @math , the sub-packetization @math satisfies the bound @math . In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any @math and @math , we presented an explicit construction of optimal-access MDS codes with sub-packetization @math . In this paper, we take up the question of reducing the sub-packetization value @math to make it to approach the lower bound. We construct an explicit family of optimal-access codes with @math , which differs from the optimal value by at most a factor of @math . These codes can be constructed over any finite field @math as long as @math , and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call group repair with optimal access . In this variation, we assume that the set of @math nodes is partitioned into @math repair groups of size @math , and require that the amount of accessed data for repair is the smallest possible whenever the @math helper nodes include all the other @math nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field @math of size @math , and also afford low-complexity encoding and decoding procedures. Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure-tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes (Dimakis et. al., 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property. Explicit constructions of such codes in the literature are only available for the cases where there are at most 3 parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math where @math The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property. Maximum distance separable (MDS) codes are optimal error-correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes. Suppose that an MDS code with @math information nodes and @math parity nodes is used to encode data in a distributed storage system. It is known that if @math out of the @math nodes are inaccessible and @math surviving (helper) nodes are used to recover the lost data, then we need to download at least @math fraction of the data stored in each of the helper nodes ( , 2010 and , 2013). If this lower bound is achieved for the repair of any @math erased nodes from any @math helper nodes, we say that the MDS code has the @math -optimal repair property. We study high-rate MDS array codes with the optimal repair property (also known as minimum storage regenerating codes, or MSR codes). Explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes, and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes. In this paper, given any @math and @math , we present two explicit constructions of MDS array codes with the @math -optimal repair property for all @math and @math simultaneously. Codes in the first family can be constructed over any base field @math as long as @math , where @math . The encoding, decoding, repair of failed nodes, and update procedures of these codes all have low complexity. Codes in the second family have the optimal access property and can be constructed over any base field @math as long as @math . Moreover, both code families have the optimal error resilience capability when repairing failed nodes. We also construct several other related families of MDS codes with the optimal repair property.