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 Singletonlike 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 Singletonlike 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 ReedSolomon 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 BerlekampJustesen 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 paritycheck polynomial of [13]. It is clear that the dimension of equals . For any codeword , we correspond it to a polynomial . If and are coprime, 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 nonnegative integer, then .
Lemma 2.3
[1, BettiSala bound] Let be nonnegative 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 nonnegative 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 nonzero codeword of with the minimum Hamming weight , and for , where and . There exists a nonzero 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 nonzero, there is at least one nonzero tuple for . Assume that is a nonzero 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 . Sincewhere . 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 ReedSolomon 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 BerlekampJustesen 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 nonzero columns whose rows are codewords in such that any nonzero columns are linearly independent over .
Proof. Assume that is a matrix over with only nonzero columns whose rows are codewords in such that any nonzero columns of are linearly independent over . Denote the index set of all nonzero 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 paritycheck 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 paritycheck matrix of , and
is a zero matrix. Then we have
, so the rows of are codewords of , and the number of nonzero columns of is . Moreover, since has locality, any nonzero 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 nonzero columns whose rows are codewords in such that any nonzero 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 nonzero columns and any columns among them are linear independent, also has only nonzero 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 nonzero 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 paritycheck 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 nonzero, 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 nonzero columns of form an matrix as following:
Since is a paritycheck matrix of , any columns of are linearly independent over . So, any columns of are linearly independent over . By Lemma 3.1 the cyclic code
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