Optimal cyclic (r,δ) locally repairable codes with unbounded length

Locally repairable codes with locality r (r-LRCs for short) were introduced by Gopalan et al. 1 to recover a failed node of the code from at most other r available nodes. And then (r,δ) locally repairable codes ((r,δ)-LRCs for short) were produced by Prakash et al. 2 for tolerating multiple failed nodes. An r-LRC can be viewed as an (r,2)-LRC. An (r,δ)-LRC is called optimal if it achieves the Singleton-type bound. Lots of works have been proposed for constructions of optimal LRCs. Among these constructions, codes with large code length and small alphabet size are of special interests. Recently, Luo et al. 3 presented a construction of optimal r-LRCs of minimum distance 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of the alphabet size) via cyclic codes. In this paper, inspired by the work of 3, we firstly construct two classes of optimal cyclic (r,δ)-LRCs with unbounded lengths and minimum distances δ+1 or δ+2, which generalize the results about the δ=2 case given in 3. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic (r,δ)-LRCs with unbounded length and larger minimum distance 2δ. Furthermore, when δ=3, we give another class of optimal cyclic (r,3)-LRCs with unbounded length and minimum distance 6.

Authors

• 10 publications
• 24 publications
• Optimal locally repairable codes of distance 3 and 4 via cyclic codes

Like classical block codes, a locally repairable code also obeys the Sin...
01/11/2018 ∙ by Yuan Luo, et al. ∙ 0

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

An (r, δ)-locally repairable code ((r, δ)-LRC for short) was introduced ...
07/29/2020 ∙ by Jing Qiu, et al. ∙ 0

• Long Optimal LRC Codes with Unbounded Localities and Unbounded Minimum Distances

A code over a finite field is called locally recoverable code (LRC) if e...
10/16/2019 ∙ by Hao Chen, et al. ∙ 0

• Improved Singleton bound on insertion-deletion codes and optimal constructions

Insertion-deletion codes (insdel codes for short) play an important role...
05/05/2021 ∙ by Bocong Chen, et al. ∙ 0

• Long Optimal or Small-Defect LRC Codes with Unbounded Minimum Distances

A code over a finite field is called locally recoverable code (LRC) if e...
10/16/2019 ∙ by Hao Chen, et al. ∙ 0

• New Constructions of Optimal Locally Repairable Codes with Super-Linear Length

As an important coding scheme in modern distributed storage systems, loc...
12/30/2020 ∙ by Xiangliang Kong, et al. ∙ 0

• Three New Infinite Families of Optimal Locally Repairable Codes from Matrix-Product Codes

Locally repairable codes have become a key instrument in large-scale dis...
09/28/2021 ∙ by Gaojun Luo, et al. ∙ 0

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1 Introduction

Motivated by applications to distributed storage, locally repairable codes (LRC) were introduced by Gopalan et al. [1], which have attracted great attention of researchers recently. Such repair-efficient codes are already used in the Hadoop Distributed File System RAID by Facebook and Windows Azure Storage [18, 19]. Let be an linear code. The -th code symbol of is said to have locality () if it can be recovered by accessing at most other symbols in , i.e., the -th code symbol can be expressed as a linear combination of other symbols. If all symbols of have locality , then is called an -LRC. Any -LRC has to satisfy the Singleton-type bound, which was proposed in [1]:

 d≤n−k−⌈kr⌉+2. (1)

When multiple node failures occur in a distributed storage system, the local recovery process for a failed node may not proceed successfully. In order to overcome this problem, Prakash et al. [2] introduced the concept of -locality, which generalize the -locality. The -th code symbol of is said to have locality ( and ), if there exists a subset such that and the punctured code has minimum distance . And is called an -LRC if all nodes of have locality . When , it is easy to see that an -LRC degenerates to an -LRC. For an linear code with -locality, Prakash et al. [2] gave the following Singleton-type bound:

 d≤n−k+1−(⌈kr⌉−1)(δ−1). (2)

An -LRC with locality (resp. ) is called optimal if it achieves the bound (2) (resp. (1)).

Lots of works have been proposed for construction of optimal LRCs ([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]). In [11], optimal -LRCs were constructed with alphabet size which is exponential in code length . A class of optimal -LRCs with length was obtained in [2] for . A breakthrough construction given in [4] produces a family of optimal -LRCs via subcodes of Reed-Solomon codes. The length of these codes can go up to the alphabet size. By employing the techniques of cyclic MDS codes [16], Chen et al. [5] obtained several classes of -ary optimal cyclic -LRCs with length . In [6], Jin et al. also constructed a family of -ary optimal -LRCs with length up to by using the automorphism group of rational function fields. By studying the algebraic structures of elliptic curves, Ma et al. [7] construct a family of -ary optimal -LRCs of length up to . One natural question is that how long can an optimal LRC be? Surprisingly, it was shown in [3] that there exist optimal cyclic -LRCs with unbounded lengths and minimum distances 3 or 4.

In this paper, we generalize the work of [3] to the -LRCs for general . Firstly, we construct two classes of optimal cyclic -LRCs with unbounded length and minimum distances and . The main results of [3] then can be seen as the case of ours. Secondly, under a slightly stronger condition, we present a construction of optimal cyclic -LRCs with unbounded length and larger minimum distance . When , with a modification of this construction, we construct another class of optimal cyclic -LRCs with unbounded length and minimum distance 6. More precisely, we have the following main results in this paper.

• (i) Suppose . Let with , then there exists a -ary optimal cyclic -LRC with length and minimum distance .

• (ii) Suppose . Let with , and , then there exists a -ary optimal cyclic -LRC with length and minimum distance .

• (iii) Suppose . Let with , and , then there exists a -ary optimal cyclic -LRC with length and minimum distance .

• (iv) Suppose

is odd and

. Let with , and , then there exists a -ary optimal cyclic -LRC with length and minimum distance .

The rest of this paper is organized as follows. In Section 2, we review some preliminaries on cyclic codes and present some basic results of cyclic -LRCs. In Section 3, we present our constructions of optimal cyclic -LRCs. We use some conclusions to end this paper in Section 4.

2 Preliminaries

In this section, we review some preliminaries on cyclic codes and present some basic results of cyclic -LRCs.

2.1 Cyclic codes

Throughout this paper, we let be a prime power and be a finite field with size . A linear code of length over is called cyclic if implies that . It is well-known that a -ary cyclic code of length can be identified with an ideal of the ring , where . Since is principal, every ideal of is generated by a monic polynomial with . is called the generator polynomial of .

Let be the order of modulo , i.e., the least number of such that . Then is a splitting field of . Let be a primitive -th root of unity. Let be a -ary -cyclic code with generator polynomial . The zeros set of is called the complete defining set of . The following lemma is a simple generalization of the well-known BCH Bound.

Lemma 1.

(Generalized BCH Bound, [15]) Let be a -ary cyclic code of length , where . Let be the generator polynomial of , and be a primitive -th root of unity. If ) has among its zeros, where is an integer and . Then the minimum distance of is at least .

2.2 Cyclic (r,δ)-LRCs

When , it was proved in [10, Theorem 10] that there is no -LRCs with achieving the bound (2). Thus, throughout this paper, we assume that and . Let be a primitive -th root of unity, where is the splitting field of .

When , Tamo et al. provided a useful condition to ensure a cyclic code has locality in [10, Proposition 3.4]. B. Chen et al. [5] then generalized their results to the cyclic -LRCs of length or . Actually, their results can be easily generalized for general with , which are presented as follows.

Lemma 2.

Suppose that , and . Let be an arithmetic progression with items and common difference , where . Consider a matrix with the rows

 h(ℓi)j=(1,ξj(r+δ−1)+ℓi,ξ2(j(r+δ−1)+ℓi),…,ξ(n−1)(j(r+δ−1)+ℓi)),

where ,

. Then all the cyclic shifts of the row vectors of weight

in the following -matrix

 V=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝10⋯0ξρℓ10⋯0(ξρℓ1)20⋯0(ξρℓ1)r+δ−20⋯010⋯0ξρℓ20⋯0(ξρℓ2)20⋯0(ξρℓ2)r+δ−20⋯0⋮⋮⋮⋮⋱⋮⋮10⋯0ξρℓδ−10⋯0(ξρℓδ−1)20⋯0(ξρℓδ−1)r+δ−20⋯0⎞⎟ ⎟ ⎟ ⎟ ⎟⎠

are contained in the row space of over .

Proof.

Note that for any and ,

Thus the row vectors of are contained in the row space of over . It is easy to see that the row space of over is closed under cyclic shifts, thus the lemma follows. ∎

Proposition 1.

Suppose that , and . Let be a cyclic code of length over with complete defining set . Let be an arithmetic progression with items and common difference , where . If contains some cosets of the group of -th roots of unity , where

 Lℓ={ξi∣i mod (r+δ−1)=ℓ},ℓ=ℓ1,ℓ2,…,ℓδ−1,

then has -locality.

Proof.

Note that

 Lℓ={ξj(r+δ−1)+ℓ}ρj=1.

Let and be the matrices defined in Lemma 2, then forms a parity-check matrix of the cyclic code . By Lemma 2, is contained in the row space of the parity check matrix . Let be the non-zero columns of , i.e.,

 V′=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1ξρℓ1(ξρℓ1)2⋯(ξρℓ1)r+δ−21ξρℓ2(ξρℓ2)2⋯(ξρℓ2)r+δ−2⋮⋮⋮⋱⋮1ξρℓδ−1(ξρℓδ−1)2⋯(ξρℓδ−1)r+δ−2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠.

Since and , . Thus for any , is not divisible by . Hence and is a parity-check matrix of an Reed-Solomon code. Then we can obtain that has -locality similarly as [5, Proposition 6]. ∎

3 Constructions

In this section, by generalizing the technique proposed in [3], we provide four classes of -ary optimal -LRCs via cyclic codes. The lengths of these codes are unbounded, i.e., lengths are independent of .

Theorem 1.

Let be a prime power and be positive integer with . Let such that . Then there exists a -ary optimal cyclic -LRC of length and minimum distance .

Proof.

Let be a primitive -th root of unity and , where . Since , , thus . Let

 g(x)=(x−1)(xρ−α)(xρ−α2)⋯(xρ−αδ−1).

Then is a polynomial over and since all roots of are -th roots of unity and they are distinct. Let be the cyclic code with generator polynomial . Then the dimension of is . Note that the set of the roots of contains where

 Lℓ={ξi∣i mod (r+δ−1)=ℓ}.

By Proposition 1, has -locality. As are roots of , the minimum distance of is at least by Lemma 1. Note that . By the bound (2),

 d≤n−(rρ−1)−(ρ−1)(δ−1)+1=δ+1.

Thus . The proof is completed. ∎

Remark 1.

[3, Theorem 1 (1)] can be seen as the case of Theorem 1.

Example 1.

Let and , then by Theorem 1, for any with , there exists an -ary optimal cyclic -LRC of length and minimum distance .

Theorem 2.

Let be a prime power and be a positive integer with . Let and such that and . Then there exists a -ary optimal cyclic -LRC of length and minimum distance .

Proof.

Let be a primitive -th root of unity and be a primitive -th root of unity, where . Since , , thus . Since , there exist integers , such that . Let . Then

 γρ=αaρ=αδ−b(r+δ−1)=αδ.

Let

 g(x)=(x−1)(x−γ)(xρ−α)(xρ−α2)⋯(xρ−αδ−1).

Then is a polynomial over and since all roots of are -th roots of unity and they are distinct. Let be the cyclic code with generator polynomial . Then the dimension of is . Note that the set of the roots of contains where

 Lℓ={ξi|i mod (r+δ−1)=ℓ}.

By Proposition 1, has -locality. Note that since . By the bound (2),

 d≤n−(rρ−2)−(ρ−1)(δ−1)+1=δ+2.

Thus to prove is optimal, we only need to show that . By contradiction, we suppose . Then there exists a nonzero polynomial with , such that . Since is a primitive -th root of unity,

 ξi≠ξj, for any 0≤i≠j≤δ.

Let

 1=(1,1,…,1),
 γ=(1,γk1,…,γkδ),

and

 ui,j=(1,ξ(i+j(r+δ−1))k1,…,ξ(i+j(r+δ−1))kδ),i=1,2,…,δ−1,j=0,1,…,ρ−1,

be the vectors in . Since and are roots of ,

 <1,c>=<γ,c>==0, (3)

for all and , where and is the canonical Euclidean inner product of . Let be the vector space spanned by the vectors ( and ) over . Since and Eq. (3), we have , i.e., any vectors of are linearly dependent. In other words, we have

Fact 1: Let be a matrix whose row vectors belong to , then .

Now, assume that among these integers , integers are divisible by and the rest integers are not divisible by . Then . Without loss of generality, we assume that . Then for any , we have

 (4)

Set

 A=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1u1,0u2,0⋮uδ−1,0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

Case (i): . By Eq. (4), we have

 ui≜ui,0+ui,1+⋯+ui,ρ−1=ρ(1,ξik1,…,ξikt,0,…,0),

for . Each . Let

 B=1ρ⎛⎜ ⎜ ⎜ ⎜⎝u1u2⋮ut+1⎞⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1ξk1…ξkt0…01ξ2k1…ξ2kt0…0⋮⋮⋱⋮⋮⋱⋮1ξ(t+1)k1…ξ(t+1)kt0…0⎞⎟ ⎟ ⎟ ⎟ ⎟⎠.

Then the first columns of form a Vandermonde matrix which is invertible since for any . We deduce that . Let

 M=(Ae).

Then

 det(M) = det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝11⋯11ξk1⋯ξkδ⋮⋮⋱⋮1ξ(δ−1)k1⋯ξ(δ−1)kδ10⋯0⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ = det⎛⎜ ⎜ ⎜ ⎜ ⎜⎝11⋯1ξk1ξk2⋯ξkδ⋮⋮⋱⋮ξ(δ−1)k1ξ(δ−1)k2⋯ξ(δ−1)kδ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ = ∏1≤i

which contradicts to the Fact 1.

Case (ii): . At this time, by Eq. (4), we have

 uδ−1=ρ(1,ξ(δ−1)k1,…,ξ(δ−1)kδ−1,0),

and

 vδ−1≜uδ−1,0−1ρuδ−1=(0,0,…,0,ξ(δ−1)kδ).

Let

 M=(Avδ−1).

Then

 det(M) = det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝11⋯111ξk1⋯ξkδ−1ξkδ⋮⋮⋱⋮⋮1ξ(δ−1)k1⋯ξ(δ−1)kδ−1ξ(δ−1)kδ00⋯0ξ(δ−1)kδ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ = ξ(δ−1)kδdet⎛⎜ ⎜ ⎜ ⎜ ⎜⎝11⋯11ξk1⋯ξkδ−1⋮⋮⋱⋮1ξ(δ−1)k1⋯ξ(δ−1)kδ−1⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ = ξ(δ−1)kδ∏0≤i

which also contradicts to the Fact 1.

Case (iii): . Recall that , and . Since , we have

 γkj=(αaρ)kjρ=(αδ−b(r+δ−1))kjρ=αkjρδ=(ξρ)kjρδ=ξδkj,

for Let

 M=(Aγ).

Then

 det(M) = det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝111⋯11ξk1ξk2⋯ξkδ1ξ2k1ξ2k2⋯ξ2kδ⋮⋮⋮⋱⋮1ξδk1ξδk2⋯ξδkδ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ = ∏0≤i

which still contradicts to the Fact 1.

In each case, it always leads to a contradiction. Thus . The proof is completed. ∎

Remark 2.

[3, Theorem 1 (2)] can be seen as the case of Theorem 2.

Example 2.

Let , and , then by Theorem 2, for any with , there exists a -ary optimal cyclic -LRC of length and minimum distance .

If we further assume that “ and ” in Theorem 2, then we can obtain an optimal cyclic -LRC with larger minimum distance as follows.

Theorem 3.

Let be a prime power and be a positive integer with . Let such that and . Then there exists a -ary optimal cyclic -LRC of length and minimum distance .

Proof.

Let be a primitive -th root of unity and be a prmitive -th root of unity, where . Since , , thus . Since , there exist integers , such that . Let . Then

 γρ=αaρ=α1−b(r+δ−1)=α.

Let

 g(x)=(x−1)δ−1∏i=1(xρ−αi)2δ−2∏j=δ(x−γj).

Note that for any and . Then is a polynomial over and since all roots of are -th roots of unity and they are distinct. Let be the cyclic code with generator polynomial . Then the dimension of is . Note that the set of the roots of contains where

 Lℓ={ξi∣i mod (r+δ−1)=ℓ}.

By Proposition 1, has -locality. Note that since . By the bound (2),

 d≤n−(rρ−δ)−(ρ−1)(δ−1)+1=2δ.

Thus to prove is optimal, we only need to prove that . By contradiction, we suppose . Then there exists a nonzero polynomial with , such that . Since is a primitive -th root of unity, we have

 ξi≠ξj, for any 0≤i≠j≤2δ−2.

Set

 1=(1,1,…,1),
 γℓ=(1,γℓk1,…,γℓk2δ−2),ℓ=δ,δ+1,…,2δ−2,

and

 ui,j=(1,ξ(i+j(r+δ−1))k1,…,ξ(i+j(r+δ−1))k2δ−2),i=1,2,…,δ−1,j=0,1,…,ρ−1,

be the vectors in . Let be the vector space spanned by the vectors and () over . Similar to the proof of Theorem 2, we have

Fact 2: Let be a matrix whose row vectors belong to , then .

Now, assume that among integers , integers are divisible by and the rest integers are not divisible by . Then . Without loss of generality, we suppose that and . Then for any , we have

 (5)

Let

 A=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1u1,0⋮uδ−1,0⎞⎟ ⎟ ⎟ ⎟ ⎟⎠

be a matrix over .

Case (i): . For , by Eq. (5), we have

 ui≜ui,0+ui,1+⋯+ui,ρ−1=ρ(1,ξik1,…,ξikt,0,…,0),

and

 vi≜ui,0−1ρui=(0,…,0,ξikt+1,…,ξik2δ−2).

Thus Recall that , and , we have

 γkj=(αaρ)kjρ=(α1−b(r+δ−1))kjρ=αkjρ=(ξρ)kjρ=ξkj,

for . Let

 M=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Av1⋮v2δ−2−tγδ⋮γt⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

Then is a matrix whose row vectors belong to , and

 det(M) = det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1