On binary locally repairable codes with distance four

Abstract

We propose a unified construction for binary locally repairable codes (LRCs) with distance four. When there is a binary linear code $[n,k,4]$ without all zero coordinates, we can construct a binary LRC $[n,k,4]$ with very good locality r. Conditions for which these LRCs attain the Cadambe-Mazumdar bound are also presented. We prove that all our LRCs with locality $r\le 3$ are optimal LRCs, and most of our LRCs with locality $r\ge 4$ can achieve the Cadambe-Mazumdar bound. Especially, if $n\le 256$, all the constructed distance optimal $[n,k,4]$ LRCs, except 17 codes, can achieve the Cadambe-Mazumdar bound. We also show that six known constructions are covered in our construction, and eight known constructions can be improved by our construction.

Introduction

Locally repairable codes are a family of erasure codes that can recover any code symbol by accessing other survived code symbols. Such codes can be used in distributed storage systems to improve repair efficiency. The concept of codes with locality was introduced by Gopalan et al. [1], Oggier and Datta [2], and Papailiopoulos et al. [3], [4]. For a q-ary linear code $\mathcal{C}={[n,k,d]}_q$, if for any $\mathbf{c}=(c_1,c_2,\cdots ,c_n)$ ∈ $\mathcal{C}$, the i-th code symbol $c_i$ can be recovered by accessing no more than $r_i$ other code symbols, $c_i$ is said to have locality $r_i$. Further, $\mathcal{C}$ has locality r if all its symbols have locality at most r, and it is denoted as $\mathcal{C}={[n,k,d;r]}_q$. If $q=2$, we write it as $\mathcal{C}=[n,k,d;r]$.

In [1], a Singleton-like bound of an ${[n,k,d;r]}_q$ was proved as:

$$d\le n-k+2-\lceil \frac{k}{r}\rceil .$$

If $r=k$, the bound (1) degenerates to the classical Singleton bound. Like the classical Singleton bound, the Singleton-like bound (1) does not take into account the cardinality of the code alphabet size q, and it is not tight in many cases. A bound taking field size into consideration was presented in [5], which is called Cadambe-Mazumdar (CM) bound. This bound says that an ${[n,k,d;r]}_q$ code satisfies

$$k\le k_{cm}=\underset{t\in {\mathbb{Z}}^{+}}{\min }\{tr+k_{opt}^{(q)}(n-t(r+1),d)\},$$

where $k_{opt}^{(q)}(n,d)$ is the largest possible dimension of a code of length n, for given field size q and minimum distance d. If $q=2$, $k_{opt}^{(q)}(n,d)$ will be written as $k_{opt}(n,d)$.

Many constructions of LRCs achieving the upper bound of (1) or (2) have been proposed in [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and references therein. Among these work, binary LRCs (BLRCs) receive much more attention, since they are easily implemented and no multiplications are needed in encoding, decoding and repair, see [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].

BLRCs with $r=2,3$ that meet the CM bound (2) have been constructed by using anti-codes [6]. Authors of [7], [8], [9], [10] proposed constructions of BLRCs with specific parameters by using cyclic codes. Authors of [11], [12], [13], [14], [15], [16], [17] discussed constructions of BLRCs from classical codes and d-optimal codes and obtained many codes meeting the CM bound (2).

In [18], the authors focused on BLRCs with $d=4$ and locality $1\le r\le 7$. Those BLRCs achieved the bound (1) when $r\in \{1,3\}$ and achieved the bound (2) when $r\in \{2,4,5,6,7\}$, all of which satisfy $(r+1)|n$. They provided a mean-time to data-loss (MTTDL) analysis on their codes and showed that $d=4$ was sufficient such that the reliability of LRCs surpassed that of the widely used 3-replication method for most practical scenarios. In [15], Rao determined the locality for binary $[n,n-m,4]$ codes with $2^{m-2}+1\le n\le 2^{m-1}$ for $4\le m\le 9$ and showed that all these BLRCs achieved the bound (2).

Hao et al. gave constructions of LRCs from parity check matrices of linear codes in [19] and [20]. In [20], BLRCs achieving the bound (1) were discussed and a class of BLRCs with d=4 and $(r+1)|n$ achieving the bound (2) was also constructed. The authors of [20] claimed that there were only four classes of BLRCs with distance 2 or 4 attaining the bound (1), yet this claim is incomplete. In [21], results of [19] and [20] were further improved, a missing class of BLRCs attaining the bound (1) in [20] was added there. Here, we list BLRCs attaining the bound (1) in [12] and [21], and known BLRCs with distance four in [20], [18] and [15].

Proposition 1 [21], [18], [12]

There are only five classes of $[n,k,d;r]$ LRCs with $2\le k\le n-2$ attaining the Singleton-like bound:

(1) $[k+k/r,k,2;r]$ for $1\le r<k$ and $r|k$;

(2) $[k+\lceil k/r\rceil ,k,2;r]$ for $1\le r<k$ and $k\ne 0(modr)$;

(3) $[2k+2,k,4;1]$ for $k\ge 2$;

(4) $[4l,3l-2,4;3]$ for $l\ge 2$;

(5) $[6,3,3;2]$, $[7,3,4;2]$, $[7,4,3;3]$ and $[8,4,4;3]$.

Proposition 2

There are the following BLRCs with distance four.

(1) ([20]) If $r\ge 1$ and $l\ge 2$, then there are $[(r+1)l,rl-\lceil lo{g}_2(r+1)\rceil ,4;r]$ BLRCs attaining the CM bound.

(2) ([15]) If $4\le m\le 9$ and $3\times 2^{m-3}+1\le n\le 2^{m-1}$, then there are $[n,n-m,4;2^{m-2}-j-1]$ BLRCs for $n=2^{m-1}-j$ and $0\le j\le 2^{m-3}$, these BLRCs achieve the CM bound.

An ${[n,k,d]}_q$ code $\mathcal{C}$ is d-optimal if there is no ${[n,k,d+1]}_q$ code. The BLRCs of Proposition 2 (2) are all d-optimal codes. According to [32], the LRCs given in Proposition 1 only include 14 d-optimal codes, namely $[2k+2,k,4;1]$ LRCs for $2\le k\le 7$, $[4l,3l-2,4;3]$ LRCs for $2\le l\le 8$ and $[7,3,4;2]$ LRC. The others are not d-optimal codes. There are some d-optimal codes in Proposition 2 (1) when r and l are chosen properly.

To determine the optimality of LRCs, we give the following definition:

Definition

An LRC $\mathcal{C}$ $=[n,k,d;r]$ is optimal if it achieves one of the following conditions:

(1) $\mathcal{C}$ attains the Singleton-like bound.

(2) $\mathcal{C}$ attains the CM bound.

(3) An $[n,k,d;\le r-1]$ LRC does not exist.

Inspired by those constructions in [18], [20] and [15], we will focus on constructing BLRCs with distance four and determine their optimality.

The contribution to this paper is three-fold. Firstly, we determine code length with which an $[n,k,4]$ code exists, then construct $[n,k,4;r]$ LRC with r as small as possible. We also show that our $[n,k,4;r]$ LRCs with $1\le r\le 3$ are all optimal BLRCs. Secondly, we give criterion for which $[n,k,4;r]$ LRCs attain the CM bound with $r\ge 4$, and prove that most of the obtained BLRCs can achieve the CM bound. Finally, for $n\le 256$, we determine all d-optimal $[n,k,4;r]$ codes that can achieve the CM bound.

The rest of this paper is organized as follows. In the next section, we begin with some preliminaries on linear codes and LRCs, and known results on judging the locality of linear codes. In Section 3, we present our main results on construction of $[n,k,4;r]$ BLRCs, and determine their optimality. Section 4 makes comparison with related results. The last section is the conclusion.