Cyclic linear binary locally repairable codes

Abstract

Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size q, i.e., the size of the code alphabet. In particular, for the binary case, only a few specific results are known by Goparaju and Calderbank. Recently, however, an upper bound on the dimension k of LRCs was presented by Cadambe and Mazumdar. The bound takes into account the length n, minimum distance d, locality r, and field size q, and it is applicable to both non-linear and linear codes.In this work, we first develop an improved version of the bound mentioned above for linear codes. We then focus on cyclic linear binary codes. By leveraging the cyclic structure, we notice that the locality of such a code is determined by the minimum distance of its dual code. Using this result, we investigate the locality of a variety of well known cyclic linear binary codes, e.g., Hamming codes and Simplex codes, and also prove their optimality with our improved bound for linear codes. We also discuss the locality of codes which are obtained by applying the operations of Extend, Shorten, Expurgate, Augment, and Lengthen to cyclic linear binary codes. Several families of such modified codes are considered and their optimality is addressed. Finally, we investigate the locality of Reed-Muller codes. Even though they are not cyclic, it is shown that some of the locality results for cyclic codes still apply.

Bounds on the rate and minimum distance of codes with availability

“…It can be seen from the corollary 5 and its proof that d ⊥ i ≤ ir + 1 for r ≥ 2, t ≥ 2. Hence using this upper bound, the bound in (45) can be wriiten as: which is tighter than the bound given in [13]. The bound given in [13] is for information symbol availability.…”

“…Hence using this upper bound, the bound in (45) can be wriiten as: which is tighter than the bound given in [13]. The bound given in [13] is for information symbol availability. But here we consider all symbol availability.…”

“…Approaches for local recovery from multiple erasures can be found in [9], [7], [10], [1], [11], [12], [13], [14], [15], [8], [16], [17]. In this paper we concentrate on codes with t availability and codes with strict t availability.…”

“…We have not defined information symbol availability code in this paper but for our purpose it suffices to note that bounds on dimension and minimum distance for t information symbol availability code also hold for codes with t availability. A field-size dependent bound is given by Huang et al in [13] for information symbol availability codes : For any linear [n, k, d] q code with t information symbol availability and hence for an (n, k, r, t) a code with minimum distance d, the dimension satisfies:…”

Locally recoverable J-affine variety codes

“…Among the different classes of codes used as good candidates for local recovering, cyclic codes and subfield-subcodes of cyclic codes play an important role, as the cyclic shifts of a recovery set provide again recovery sets [1,10,12,17]. In this article we continue this line of research by using the very general language of affine variety codes.…”