Applications of polymatroid theory to distributed storage systems

Abstract

Abstract-In this paper, a link between polymatroid theory and locally repairable codes (LRCs) is established. The codes considered here are completely general in that they are subsets of A n , where A is an arbitrary finite set. Three classes of LRCs are considered, both with and without availability, and for both information-symbol and all-symbol locality. The parameters and classes of LRCs are generalized to polymatroids, and a generalized Singelton bound on the parameters for these three classes of polymatroids and LRCs is given. This result generalizes the earlier Singleton-type bounds given for LRCs. Codes achieving these bounds are coined perfect, as opposed to the more common term optimal used earlier, since they might not always exist. Finally, new constructions of perfect linear LRCs are derived from gammoids, which are a special class of matroids. Matroids, for their part, form a subclass of polymatroids and have proven useful in analyzing and constructing linear LRCs.

Matroid Theory and Storage Codes: Bounds and Constructions

“…We will briefly discuss the connection between polymatroids and codes in Section5. Singleton-type bounds for polymatroids were derived in [46], and polymatroid theory for its part seems to be particularly suitable for proving such bounds for general LRCs. In Section 5 we will also review Singleton-type bounds for polymatroids and general codes with availability and hierarchy.…”

“…In addition, the parameters (n, k, d, r, δ ) for linear LRCs were generalised to matroids, and new results for both matroids and linear LRCs were given therein. Even more generally, the parameters (n, k, d, r, δ ,t) were generalised to polymatroids, and new results for polymatroids, matroids and both linear and nonlinear LRCs over arbitrary finite alphabets were derived in [46]. Similar methods can be used to bound parameters of batch codes [49], as discussed in Chapter 16.…”

“…In particular, we are focusing on how the parameters of a LRC can be analysed using the lattice of cyclic flats of an associated matroid, and on a construction derived from matroid theory that provides us with linear LRCs. The matroidal results on LRCs reviewed here are mostly taken from [47,46,10].…”

Batch and PIR Codes and Their Connections to Locally Repairable Codes

“…By building on the ideas in the previous works, the authors derive a variation of the bound (3) for the model under consideration. Other related works, for example, include [6,19,21,29,35,36] and the references therein.…”

Bounds on the maximal minimum distance of linear locally repairable codes

“…Consequently, the parameters (n, k, d, r, δ) were generalized to matroids and the bound (1) was proven to also hold for all matroids, which is nontrivial since not all matroids are induced by almost affine codes. An even more general Singleton bound was given for polymatroids in [11], motivated by the fact that all general LRCs induce a polymatroid.…”