We introduce a new family of erasure codes, called group decodable code (GDC), for distributed storage system. Given a set of design parameters ; ; k; t , where k is the number of information symbols, each codeword of an ( ; ; k; t)-group decodable code is a t-tuple of strings, called buckets, such that each bucket is a string of symbols that is a codeword of a [ ; ] MDS code (which is encoded from information symbols). Such codes have the following two properties: (P1) Locally Repairable: Each code symbol has locality ( ; - + 1). (P2) Group decodable: From each bucket we can decode information symbols. We establish an upper bound of the minimum distance of ( ; ; k; t)-group decodable code for any given set of ; ; k; t ; We also prove that the bound is achievable when the coding field F has size |F| > n-1 k-1.
In @cite_12 , the concept of @math -locality was defined, which captures the property that there exist @math pairwise disjoint local repair sets for a code symbol. An upper bound on the minimum distance for @math linear codes with information @math -locality was derived, and codes that attain this bound was constructed for the length @math . However, for @math , it is not known whether there exist codes attaining this bound. Upper bounds on the rate and minimum distance of codes with all-symbol @math -locality was proved in @cite_13 . However, no explicit construction of codes that achieve this bound was presented. It is still an open question whether the distance bound in @cite_13 is achievable.
@cite_13 @cite_12
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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. 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. In this work, we present a new upper bound on the minimum distance d of linear locally repairable codes (LRCs) with information locality and availability. The bound takes into account the code length n, dimension k, locality r, availability t, and field size q. We use tensor product codes to construct several families of LRCs with information locality, and then we extend the construction to design LRCs with information locality and availability. Some of these codes are shown to be optimal with respect to their minimum distance, achieving the new bound. Finally, we study the all-symbol locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check (LDPC) codes. We also investigate their optimality using the new bound.