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.
Locality was identified as a repair cost metric for distributed storage systems independently by Oggier @cite_13 , Gopalan @cite_9 and PaPailiopoulos @cite_12 using different terms. In @cite_9 , Gopalan introduced the concept of symbol locality of linear codes and established a tight bound for the redundancy in terms of the message length, the distance, and the locality of information coordinates. A generalized concept, i.e., @math locality, was addressed by Prakash @cite_2 . It was proved in @cite_2 that the minimum distance @math of an @math linear code @math is upper bounded by where @math and @math are the length and dimension of @math respectively. It was also proved that a class of codes known as pyramid codes @cite_6 achieve this bound. Since an @math code is also an @math code, ) also presents an upper bound for the minimum distance of @math codes.
@cite_9 @cite_6 @cite_2 @cite_13 @cite_12
2018102393 1993830711 1969158823 1678309266
In distributed storage systems, erasure codes with locality (r ) are preferred because a coordinate can be locally repaired by accessing at most (r ) other coordinates which in turn greatly reduces the disk I O complexity for small (r ) . However, the local repair may not be performed when some of the (r ) coordinates are also erased. To overcome this problem, we propose the ((r, )_ c ) -locality providing ( -1 ) nonoverlapping local repair groups of size no more than (r ) for a coordinate. Consequently, the repair locality (r ) can tolerate ( -1 ) erasures in total. We derive an upper bound on the minimum distance for any linear ([n,k] ) code with information ((r, )_ c ) -locality. Then, we prove existence of the codes that attain this bound when (n k(r( -1)+1) ) . Although the locality ((r, ) ) defined by provides the same level of locality and local repair tolerance as our definition, codes with ((r, )_ c ) -locality attaining the bound are proved to have more advantage in the minimum distance. In particular, we construct a class of codes with all symbol ((r, )_ c ) -locality where the gain in minimum distance is ( ( r ) ) and the information rate is close to 1. Consider a linear [n,k,d]q code C. We say that the ith coordinate of C has locality r , if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large distance, and low locality for parity coordinates. In this paper, we carry out an in-depth study of the relations between these parameters. We establish a tight bound for the redundancy n-k in terms of the message length, the distance, and the locality of information coordinates. We refer to codes attaining the bound as optimal. We prove some structure theorems about optimal codes, which are particularly strong for small distances. This gives a fairly complete picture of the tradeoffs between codewords length, worst case distance, and locality of information symbols. We then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes. Using our structure theorem, we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings. We prove that there is a tradeoff between having good locality and the ability to correct erasures beyond the minimum distance. Repair locality is a desirable property for erasure codes in distributed storage systems. Recently, different structures of local repair groups have been proposed in the definitions of repair locality. In this paper, the concept of regenerating set is introduced to characterize the local repair groups. A definition of locality @math (i.e., locality @math with repair tolerance @math ) under the most general structure of regenerating sets is given. All previously studied locality turns out to be special cases of this definition. Furthermore, three representative concepts of locality proposed before are reinvestigated under the framework of regenerating sets, and their respective upper bounds on the minimum distance are reproved in a uniform and brief form. Additionally, a more precise distance bound is derived for the square code which is a class of linear codes with locality @math and high information rate, and an explicit code construction attaining the optimal distance bound is obtained. The @math th coordinate of an @math code is said to have locality @math and availability @math if there exist @math disjoint groups, each containing at most @math other coordinates that can together recover the value of the @math th coordinate. This property is particularly useful for codes for distributed storage systems because it permits local repair and parallel accesses of hot data. In this paper, for any positive integers @math and @math , we construct a binary linear code of length @math which has locality @math and availability @math for all coordinates. The information rate of this code attains @math , which is always higher than that of the direct product code, the only known construction that can achieve arbitrary locality and availability.