This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters @math of LRCs are generalized to matroids, and the matroid analogue of the generalized Singleton bound in [P. , "On the locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes, that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters @math and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given in [W. , "Optimal locally repairable codes," IEEE J. Sel. Areas Comm.].
Recently, the present authors have studied locally repairable codes with all-symbol locality @cite_20 . Methods to modify already existing codes were presented and it was shown that with high probability, a certain random matrix will be a generator matrix for a locally repairable code with a good minimum distance. Constructions were given for three infinite classes of optimal vector-linear locally repairable codes over an alphabet of small size. The present paper extends and deviates from this work by studying the combinatorics of LRCs in general and relating LRCs to matroid theory. This allows for the derivation of fundamental bounds for matroids and linear and almost affine LRCs, as well as for the characterization of the matroids achieving this bound.
@cite_20
2963963730 1902314736 2950599147 1639538057
This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters @math of LRCs are generalized to matroids, and the matroid analog of the generalized singleton bound by for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters @math and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given by In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. It is also shown that, with high probability, a random matrix with a few extra columns guaranteeing the locality property is a generator matrix for a locally repairable code with a good minimum distance. The proof of the result provides a constructive method to find locally repairable codes. Finally, constructions of three infinite classes of optimal vector-linear locally repairable codes over a small alphabet independent of the code size are given. 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. 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.