In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan , in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Tur 'an graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from @math erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.
Explicit constructions of optimal codes with all-symbol locality for the single erasure case are provided in @cite_2 , @cite_5 , respectively based on Gabidullin maximum rank-distance and Reed-Solomon codes. Families of codes with all-symbol locality with small alphabet size (low field size) are constructed in @cite_13 . Locality in the context of non-linear codes is considered in @cite_16 . Codes with local regeneration are considered in @cite_10 , @cite_7 , @cite_14 . Studies on the implementation and performance evaluation of codes with locality can be found in @cite_1 @cite_12 .
@cite_13 @cite_14 @cite_7 @cite_1 @cite_2 @cite_5 @cite_16 @cite_10 @cite_12
2967056459 2951572681 2155418912 2018102393
In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated with small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by , in which recovery from a single erasure is considered. By studying the generalized Hamming weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Turan graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from 2 erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case. In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan , in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Tur 'an graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from @math erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case. Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n,k,d] linear code having all-symbol (r,δ)-locality, denoted as (r,δ)_a, is considered optimal if it has the actual highest minimum distance of any code of the given parameters n,k,r and δ. A minimum distance bound is given in . The existing results on the existence and the construction of optimal (r, δ)_a linear codes are limited to only two small regions within this special case, namely, i) m=0 and ii) m≥ (v+δ-1)>(δ-1) and δ=2, where m=n mod(r+δ-1) and v=k mod r. This paper investigates the properties and existence conditions for optimal (r,δ)_a linear 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 (r,δ)_a linear code can achieve the minimum distance bound in . For four other cases the optimal (r,δ)_a codes are shown to exist over a field of size q≥( n k-1 ), deterministic constructions are proposed. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on optimal (r,δ)_a linear codes, leaving only two cases in which the construction of optimal codes are not yet known. 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.