New Constructions of Optimal Cyclic (r,δ) Locally Repairable Codes from Their Zeros

07/29/2020 ∙ by Jing Qiu, et al. ∙ 0

An (r, δ)-locally repairable code ((r, δ)-LRC for short) was introduced by Prakash et al. <cit.> for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of r-LRCs produced by Gopalan et al. <cit.>. An (r, δ)-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. <cit.> generalized the construction of cyclic r-LRCs proposed by Tamo et al. <cit.> and constructed several classes of optimal (r, δ)-LRCs of length n for n | (q-1) or n | (q+1), respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of <cit.>, this paper first characterizes (r, δ)-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic (r, δ)-LRCs of length n for n | (q-1) or n | (q+1), respectively from the product of two sets of zeros. Our constructions include all optimal cyclic (r,δ)-LRCs proposed in <cit.>, and our method seems more convenient to obtain optimal cyclic (r, δ)-LRCs with flexible parameters. Moreover, many optimal cyclic (r,δ)-LRCs of length n for n | (q-1) or n | (q+1), respectively such that (r+δ-1)∤ n can be obtained from our method.

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1 Introduction

Let be a finite field with size and an linear code over . The -th code symbol of is said to have -locality if it can be recovered by accessing at most other code symbols in , i.e., the -th code symbol can be expressed as a linear combination of other code symbols. If all the code symbols of have locality , then is called an locally repairable code (-LRC for short). This concept was introduced firstly by Gopalan et al. [5] for application of coding techniques to distributed storage systems. It was proved in [5, 18] that the minimum distance of an -LRC  is upper bounded by

(1)

This bound is called the Singleton-like bound for LRCs. The linear codes meeting the above bound (1) are called optimal -LRCs.

In order to deal with the situation that multiple node failures occur in a distributed storage system, Prakash et al. [14] introduced the concept of -locality of linear codes, where , which generalized the notion of -locality. The -th code symbol of is said to have -locality (), if there exists a subset such that , and the punctured code has the minimum distance . The code is said to have -locality or be an -LRC if all the code symbols have -localities. A Singleton-like bound for the minimum distance of an -LRC is given as follows [14]:

(2)

Linear codes meeting this bound (2) are called optimal -LRCs. Note that the notion of -locality is a special case of the notion of -locality for . In this case, the minimum distance bound (2) becomes to the bound (1).

In the last decade, many constructions of optimal LRCs have been proposed, for example see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22] and references therein. A breakthrough construction of optimal LRCs was proposed by Tamo and Barg in [18] via a generalization of the classical construction of Reed-Solomon codes. Along this way, Tamo et al. [19, 20] constructed a family of optimal cyclic -LRCs of lengths and its factors in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Then their method was generalized to the -locality case by Chen et al.[2] and they also constructed many optimal cyclic -LRCs with lengths and its factors by application of Berlekamp-Justesen codes. Very recently, many authors focused on constructions of optimal LRCs with unbounded length, for example see [12, 4, 11, 17, 9]. Among all known constructions of optimal cyclic -LRCs, the condition of has to be hold. To the best of our knowledge, there is no result about the existence or construction of optimal cyclic -LRCs such that .

Inspired by the work of [2, 3], this paper aims to construct optimal cyclic -LRCs in terms of the product of two sets of zeros of cyclic codes. First, we characterize -locality of cyclic codes from their zeros and show that a cyclic code has -locality if its complete defining set contains a product of two sets of zeros (see Theorem 3.3). By application of this result we propose several constructions of optimal cyclic -LRCs of length for or , respectively from the product of two sets of zeros of cyclic codes. All optimal cyclic -LRCs proposed in [2, 3] can be reconstructed from our method. It seems more convenient to obtain optimal cyclic -LRCs with flexible parameters. Moreover, many optimal cyclic -LRCs of length for or , respectively such that can be obtained from our constructions.

The rest of the paper is organized as follows. In Section 2, we review some preliminaries on cyclic codes and some known constructions of optimal cyclic -LRCs. Section 3 characterizes -locality of cyclic codes from their zeros. In Section 4, we construct optimal cyclic -LRCs of lengths and its factors in terms of the product of two sets of zeros. Section 5 constructs optimal cyclic -LRCs of lengths and its factors in terms of the product of two sets of zeros. Finally, Section 6 concludes this paper.

2 Preliminaries

2.1 Cyclic codes and their complete defining sets

An cyclic code over the finite field is a -dimensional linear subspace of satisfying the condition that whenever . It is well known that any cyclic code of length over corresponds to an ideal of , and can be expressed as , where is a monic polynomial over and . The is called the generator polynomial and is referred to as the parity-check polynomial of  [13]. It is clear that the dimension of equals . For any codeword , we correspond it to a polynomial . If and are co-prime, then is uniquely determined by the set of its roots. The zeros of are also called the zeros of , and is called the complete defining set of , where is the -th primitive root of unity in some extension of .

In the next sections, adopt the following notation unless otherwise stated:

  • is a power of a prime and is a finite field of size .

  • is a positive integer with , is the order of modulo and is the extension of with degree .

  • is a primitive -th root of unity in and is the set of all -th roots of unity.

  • is the set , where are subsets of .

The operation of multiplying by divides the integers modulo into sets called -cyclotomic cosets modulo . For an integer with , the -cyclotomic coset of modulo is defined by

where is the smallest positive integer such that . Let be a cyclic code over of length with the complete defining set for some . Then the set of exponents of such that , i.e., is a union of some -cyclotomic cosets modulo , since the generator polynomial of is a monic divisor polynomial over of .

Definition 2.1

A subset of is called a consecutive set of length if a primitive -th root of unity and an exponent exist such that .

The following lemmas are useful to establish our main results in this paper.

Lemma 2.2

[13, BCH bound] Let be an cyclic code over and a primitive -th root of unity. If the complete defining set of contains the following set

where is a positive integer with and is a non-negative integer, then .

Lemma 2.3

[1, Betti-Sala bound] Let be non-negative integers with . Let be an cyclic code over and a primitive -th root of unity. If the complete defining set of contains the following set,

then .

As the generalization of the BCH bound, the minimum distance bound on cyclic codes proposed in [1] can be generalized to the following case.

Lemma 2.4

Let be non-negative integers with . Let be an cyclic code over and a primitive -th root of unity. If the complete defining set of contains the following set,

where , then .

Proof. Let . Since , is also a primitive -th root of unity. So, for some integer . Then the complete defining set contains the following set:

The result follows from Lemma 2.3.

The above lemmas recall the results on the bound of the minimum distance of cyclic codes from their zeros. Next, we further characterize the minimum distance of some cyclic codes in terms of their complete defining sets.

Lemma 2.5

Let be an cyclic code over with the complete defining set , then the minimum distance of the dual code of is equal to the minimum distance of the cyclic code with the complete defining set .

Proof. Let denote the generator polynomial of and its zero set. Then the set is a union of some -cyclotomic cosets modulo . It is easy to see that the set is also a union of some -cyclotomic cosets modulo . Let denote the cyclic code of length over with the complete defining set . Then the generator polynomial of is . Let denote the dual of and it can be generated by .

For any codeword there exists such that . So,

(3)

Let , then . So,

(4)

Comparing (3) and (4) we have for . In other words, for a codeword , there always exists a codeword , and vice versa. This completes the proof.

Below we give an example to illustrate Lemma 2.5.

Example 2.6

Let , and be a primitive -th root of unity. Let and be the cyclic codes over with the complete defining sets and , respectively. Magma verifies that , where and denote the minimum distance of and the dual of , respectively. The experiment result is consistent with Lemma 2.5.

Proposition 2.7

Let denote the cyclic code of length over with the complete defining set . Let denote the cyclic code of length  over with the same complete defining set , i.e., is generated by the generator polynomial of over . Then , where and denote the minimum distance of and , respectively.

Proof. Since is the subfiled subcode of , . We only need to show . Let with degree , which generate over and over , respectively, i.e.,

Since is a cyclic code over , is a polynomial over , and assume that .

Suppose is a non-zero codeword of with the minimum Hamming weight , and for , where and . There exists a non-zero polynomial such that . So,

(5)

Let be a basis of over . Each coefficient of can be represented as follows:

(6)

where for . From assumption and equation (5), for we have

(7)

Substituting in (6) into (7) we get

where . So,

(8)

Since is non-zero, there is at least one non-zero -tuple for . Assume that is a non-zero tuple and set . From (8) we find a codeword such that for . So, . This completes the proof.

Proposition 2.8

Let be a positive integer with and be a cyclic code over with the complete defining set . If contains a coset of a subgroup of with order  and contains a consecutive set of length , then the minimum distance of the dual of is exact .

Proof. Let and denote the dual of and the minimum distance of , respectively. Let and be the subgroup of of order and for some with . Consider an matrix as follows:

It is clear that each vector in the row space of

is in . Since

where . Adding up all rows in we get a codeword in with Hamming weight . On the other hand, there exists a consecutive set of length in . By Lemma 2.5 and Lemma 2.2, . So, .

2.2 Some known constructions of optimal cyclic -LRCs

Tamo et al.[19, 20] constructed a class of optimal cyclic -LRCs with lengths and its factors based on Reed-Solomon codes. Immediately, Chen et al. [2] generalized this construction to the case of cyclic -LRCs. Moreover, they firstly constructed several classes of cyclic -LRCs with lengths and its factors by application of Berlekamp-Justesen codes. In this section, we recall the construction of optimal cyclic -LRCs with lengths and its factors in [2], and the case of lengths and its factors is referred to [2, 3].

Lemma 2.9

[2, Proposition 6] Let be positive integers such that , and be a cyclic code of length over with the complete defining set . Let be an arithmetic progression with items and the common difference , where . If contains some cosets of the group of -th roots of unity , where , and , then has -locality.

Lemma 2.10

[2, Construction 7] Let be positive integers such that . Let be a primitive -th root of unity, where . Let be an arithmetic progression with items and the common difference , where . Suppose and let . Consider the following sets of elements of :

and , where . Then the cyclic code with the complete defining set is an optimal cyclic -LRC with length , dimension and minimum distance .

3 -locality of cyclic codes

It is known that any cyclic code over has -locality, where , and is the minimum distance of the dual of . In this section, we generalize this result and characterize the -locality of cyclic codes in terms of their zeros. To this end, we first analyze the -locality of cyclic codes from their parity check matrices.

Lemma 3.1

Let be positive integers and an cyclic code over . Then has -locality if and only if there exists a matrix over with only non-zero columns whose rows are codewords in such that any non-zero columns are linearly independent over .

Proof. Assume that is a matrix over with only non-zero columns whose rows are codewords in such that any non-zero columns of are linearly independent over . Denote the index set of all non-zero columns of by and . Let denote the punctured code of over the coordinate set . If we can show that , then has -locality for the coordinates in . Since is a cyclic code, each coordinate of the codeword has -locality. Let denote the linear code such that is the parity-check matrix. Let denote the submatrix consisting of columns of whose support is . Then . It is easy to show that . So, .

Conversely, let be a recover set for some coordinate and , where . Let be the generator matrix of , then generates . Let be the matrix such that is the parity-check matrix of , and

is a zero matrix. Then we have

, so the rows of are codewords of , and the number of non-zero columns of is . Moreover, since has -locality, any non-zero columns of are linearly independent over .

In general, the zeros of a cyclic code of length  over are in the extension of , where  is the order of modulo . To characterize the -locality of cyclic codes in terms of their zeros, we still need the following lemma.

Lemma 3.2

Let be an cyclic code over and denote the subfield subcode of over , where is the order of modulo . If has -locality, then also has -locality.

Proof. Since has -locality, from Lemma 3.1, there exists a matrix over with only non-zero columns whose rows are codewords in such that any non-zero columns are linearly independent over , where denote the dual of . Let denote the column vector in corresponding to for . Then is a matrix over . For each codeword , we have . So, each row of is a codeword of the dual of . Since has only non-zero columns and any columns among them are -linear independent, also has only non-zero columns and any columns among them are -linear independent. By Lemma 3.1, has -locality.

Theorem 3.3

Let be the order of modulo , and be two subsets of such that , where and denote the minimum distance of the dual of the cyclic code of length  over with the complete defining set and the cyclic code of length  over with the complete defining set , respectively. If the complete defining set of a cyclic code of length  over contains , then has -locality.

Proof. Let be the cyclic code of length over generated by the generator polynomial of over . It is easy to see that is the subfield subcode of . By Lemma 3.2 we only need to show that has -locality.

Let . Let and denote the dual of the cyclic code of length  over with the complete defining set and the minimum distance of , respectively. Set

It is clear that is the generator matrix of . Let denote a non-zero codeword in with the minimum weight, and be the coefficients of each row of respectively in the linear combination of expression of . If we set , then

and the weight of is .

Let and denote the cyclic code of length  over with the complete defining set . Then has the following parity-check matrix:

For any , consider the following matrix associated with :

Using the same coefficients as in the expression of to make a linear combination of rows of we get

Since the powers of are non-zero, shares the same support with , whose index set is denoted by , where .

Let be a matrix whose -th row is for , which comes from the row space of , where is an matrix from the set as that is from . It is clear that each row of is a codeword of the dual of . All non-zero columns of form an matrix as following:

Since is a parity-check matrix of , any columns of are linearly independent over . So, any columns of are linearly independent over . By Lemma 3.1 the cyclic code