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.
provides background on generalized Hamming weights (GHW). Our formulation and approach to the problem are outlined in . An important connection between the @math -cores of @cite_0 and GHW is made in . The upper bound on @math and optimal code constructions can be found in Sections and respectively. The final section, presents the analogous @math bound for the single-erasure case. The proofs of most statements appear in the Appendix.
@cite_0
2967056459 2951572681 2114197496 2104812861
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. An optimal family of array codes over GF(q) for correcting multiple phased burst errors and erasures, where each phased burst corresponds to an erroneous or erased column in a code array, is introduced. As for erasures, these array codes have an efficient decoding algorithm which avoids multiplications (or divisions) over extension fields, replacing these operations with cyclic shifts of vectors over GF(q). The erasure decoding algorithm can be adapted easily to handle single column errors as well. The codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array, thus generalizing a previously known construction for the special case of two erasures. Algebraically, they can be interpreted as Reed-Solomon codes. When q is primitive in GF(q), the resulting codes become (conventional) Reed-Solomon codes of length P over GF(q sup p-1 ), in which case the new erasure decoding technique can be incorporated into the Berlekamp-Massey algorithm, yielding a faster way to compute the values of any prescribed number of errors. > Let F sub q denote the finite field GF(q) and let h be a positive integer. MDS (maximum distance separable) codes over the symbol alphabet F sub q sup b are considered that are linear over F sub q and have sparse ("low-density") parity-check and generator matrices over F sub q that are systematic over F sub q sup b . Lower bounds are presented on the number of nonzero elements in any systematic parity-check or generator matrix of an F sub q -linear MDS code over F sub q sup b , along with upper bounds on the length of any MDS code that attains those lower bounds. A construction is presented that achieves those bounds for certain redundancy values. The building block of the construction is a set of sparse nonsingular matrices over F sub q whose pairwise differences are also nonsingular. Bounds and constructions are presented also for the case where the systematic condition on the parity-check and generator matrices is relaxed to be over F sub q , rather than over F sub q sup b .