Binary Linear 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 results are known.In this work, we present an upper bound on the minimum distance d of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length n, dimension k, locality r, availability t, and field size q. Then, we study binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed-Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared to other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, availability of LRCs is studied. We investigate the 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 show the construction of a long LRC with availability from a short one-step majority-logic decodable code.

Generalized construction of cyclic (<i>r</i>, <i>t</i>)-locally repairable codes using trace function

“…It can achieve high availability with a guaranteed lower bound on the minimum distance shown in Theorem 2. The obtained code is a one-step majority-logic decodable code, whose usefulness as an ðr; tÞ LRC has already been pointed out in [4]. However, unlike the construction method developed in [4], the parity check matrix of the obtained code does not have a tensor product structure.…”

“…t þ 1, note that there are t ¼ eðm À þ 1Þ polynomials that contain the common term x q À1 as shown in (1). Each of these polynomials corresponds to a row in a parity check matrix of a one-step majority-logic decodable code [4]. It is well-known that the minimum distance d of such a majority-logic decodable code is at least t þ 1 [7].…”

“…Generally, it requires a large amount of computation to evaluate locality r and availability t for a given linear code. Huang, Yaakobi, Uchikawa, and Siegel [4] showed that a class of cyclic LRCs offers an efficient way to evaluate locality r and availability t. Wang, Zhang, and Lin [5] introduced p-ary cyclic ðr; tÞ-LRCs using the trace function over a finite field [6], where p is a prime. However, for a fixed r, it is impossible to modify code length n and dimension k, which leads to the lack of flexibility in code design.…”

Erasure coding for distributed storage: an overview

“…Classical bounds on coding theory can be applied to this shortened code, to yield "lifted" bounds on the parent code having locality. This shortening approach, presented for the first time in [109], has since been employed in subsequent papers in the literature, see [110], [111].…”

“…This bound is shown to be tighter than (21) for some cases including 5 ≤ d min ≤ 8 for n large. In [110], the authors employ the shortening approach to derive an alphabet-size-dependent bound on the minimum distance and dimension of codes having IS locality. An example comparison of the bounds on dimension for linear LR codes in (22), (21) and the Hamming-bound based bound in [113] is presented in Table III.…”