Optimal locally repairable codes of distance 3 and 4 via cyclic codes

01/11/2018 ∙ by Yuan Luo, et al. ∙ Shanghai Jiao Tong University Nanyang Technological University 0

Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code optimal if it achieves the Singleton-type bound). In the breakthrough work of TB14, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in TB14 are upper bounded by the code alphabet size q. Recently, it was proved through extension of construction in TB14 that length of q-ary optimal locally repairable codes can be q+1 in JMX17. Surprisingly, BHHMV16 presented a few examples of q-ary optimal locally repairable codes of small distance and locality with code length achieving roughly q^2. Very recently, it was further shown in LMX17 that there exist q-ary optimal locally repairable codes with length bigger than q+1 and distance propositional to n. Thus, it becomes an interesting and challenging problem to construct new families of q-ary optimal locally repairable codes of length bigger than q+1. In this paper, we construct a class of optimal locally repairable codes of distance 3 and 4 with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen.

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1 Introduction

Due to applications to distributed storage systems, locally repairable codes have recently attracted great attention of researchers [5, 4, 12, 13, 7, 8, 3, 11, 14, 15, 1]. A local repairable code is nothing but a block code with an additional parameter called locality. For a locally repairable code of length with information symbols and locality (see the definition of locally repairable codes in Section 2.1), it was proved in [4] that the minimum distance of is upper bounded by

(1)

The bound (1) is called the Singleton-type bound for locally repairable codes and was proved by extending the arguments in the proof of the classical Singleton bound on codes. In this paper, we refer an optimal locally repairable code to a block code achieving the bound (1).

1.1 Known results

The early constructions of optimal locally repairable codes gave codes with alphabet size that is exponential in code length (see [6, 13]. There was also an earlier construction of optimal locally repairable codes given in [12] with alphabet size comparable to code length. However, the construction in [12] only produced a specific value of the length , i.e., . Thus, the rate of the code is very close to . There are also some existence results given in [12] and [14] with less restriction on locality . But the results in both the papers require large alphabet size which is an exponential function of the code length.

A recent breakthrough construction given in [14] makes use of subcodes of Reed-Solomon codes. This construction produces optimal locally repairable codes with length linear in alphabet size although the length of codes is upper bounded by alphabet size. The construction in [14] was extended via automorphism group of rational function fields by Jin, Ma and Xing [7] and it turns out that there are more flexibility on locality and the code length can go up to , where is the alphabet size.

Based on the classical MDS conjecture, one should wonder if -ary optimal locally repairable codes can have length bigger than . Surprisingly, it was shown in [2] that there exist -ary optimal locally repairable codes of length exceeding . Although [2] produced a few optimal locally repairable codes with specific parameters, it paves a road for people to continue search for such optimal codes. Very recently, it was shown in [8] via elliptic curves that there exist -ary optimal locally repairable codes with length bigger than and distance proportional to length .

1.2 Our result

In this paper, by carefully choosing generator polynomials, we can show that there exist optimal locally repairable codes of distance and that are cyclic. One feature of these codes is that their lengths are unbounded, i.e., lengths are independent of the code alphabet sizes. More precisely, we have the following main result in this paper.

Theorem 1.1.

Let be a prime power. Assume that is a positive integer with . Then

  • there is a -ary optimal cyclic locally repairable code with locality if and ; and

  • there is a -ary optimal cyclic locally repairable code with locality if , and divides .

Remark 1.
  • The conditions in Theorem 1.1 are easy to be satisfied and the length is unbounded.

  • Indeed, parameters given in Theorem 1.1 satisfy the equality of (1). Let us verify this only for part (i) of Theorem 1.1. We have

1.3 Open problems

In view of the known results and our result in this paper, all known optimal -ary locally repairable codes with length and minimum distance satisfy either (i) is small and is unbounded; or (ii) is proportional to and is linear in . One natural question is

Open problem 1.

Are there optimal -ary locally repairable codes with length much bigger than (for instance ) and minimum distance proportional to ?

The other open problem is the following.

Open problem 2.

Are there optimal -ary locally repairable codes with unbounded length for every constant .

1.4 Organization of the paper

The paper is organized as follows. In Section 2, we provide some preliminaries on locally repairable codes and cyclic codes. In Section 3, we present proof of Theorem 1.1. In addition, we also give a construction of optimal -ary locally repairable codes of length and distance .

2 Preliminaries

In this section, we present some preliminaries on locally repairable codes and cyclic codes.

2.1 Locally repairable codes

Informally speaking, a block code is said with locality if every coordinate of a given codeword can be recovered by accessing at most other coordinates of this codeword. Precisely speaking, a locally repairable code with locality is given as follows.

Definition 1.

Let be a -ary block code of length . For each and , define . For a subset , we denote by the projection of on . Then is called a locally repairable code with locality if, for every , there exists a subset with such that and are disjoint for any .

Apart from the usual parameters: length, rate and minimum distance, the locality of a locally repairable code plays a crucial role. In this paper, we always consider locally repairable codes that are linear over . Thus, a -ary locally repairable code of length , dimension , minimum distance and locality is said to be an -locally repairable code with locality .

2.2 Cyclic codes

Cyclic codes are well known and understood in the coding community. We briefly introduce them and list some useful facts on cyclic codes in this subsection. A -ary cyclic code is identified with an ideal of the ring . As every ideal of is a principal ideal generated by a divisor of , we can just write . The dimension of is . Furthermore, is a codeword of if and only if is divisible by . In other words, if are all roots of (where stands for algebraic closure of ), then belongs to if and only if for all .

Let for a divisor of . Put . Let be the reciprocal polynomial of . Then the Euclidean dual code of is . We recall some facts about cyclic codes without giving proof. The reader may refer to the books [10, 9].

Lemma 2.1.
  • Let be a divisor of , then the codes and are equivalent.

  • Let a -ary cyclic code of length for a divisor of and assume that are roots of . If has a nonzero codeword of weight at most , then the determinant

    is zero.

  • Let be a -ary cyclic code of length for a divisor of . Let be an th primitive root of unity. If there exist an integer and a positive integer such that are roots of , then has minimum distance at least .

Let denote the symmetric group of length . An automorphism of a -ary block code is a permutation satisfying that whenever . All automorphisms form a subgroup of , denoted by . It is called the automorphism group of . The code is called transitive if, for any , there exists an automorphism such that .

If is the cyclic code, then contains the subgroup generated by the cyclic shift . Thus, is transitive.

3 Constructions

In this section, we first give a general result on locality for transitive codes and then apply it to the proof of Theorem 1.1. In addition, we also present a construction of -ary locally repairable code with locality .

3.1 A general result

Lemma 3.1.

If is a -ary transitive linear code with dual distance , then has locality .

Proof.

Let be a codeword of with the Hamming weight . Let denote the support of . Then . Let be a subset of such that and . Choose an element . Then for every codeword , we have . This gives . This implies that can be repaired by . Now for any , there exists an automorphism such that . This implies that

Hence, . The proof is completed. ∎

Example 3.2.

Let be a prime power. Let be positive integers with , and . We present a -ary optimal cyclic locally repairable code of length minimum distance and locality for any . Although such a code was already given in [14], we provide different view for this code. Such an observation finally leads to discovery of optimal locally repairable codes with unbounded length given in Theorem 1.1. Let be an th primitive root of unity. For a positive integer with , can be uniquely written as for some integer and integer .

Case 1. .

Let

It is obvious that since all roots of are the th roots of unity and they are distinct. We denote by the cyclic code with generator polynomial . Due to the fact that contains the roots , it follows from Lemma 2.1(i) that has the minimum distance at least . Moreover, the dimension of is . Thus, we have

Thus, to show that satisfies the Singleton-type bound (1), it remains to show that the locality of is . By Lemma 3.1, it is sufficient to the dual distance of is at most . By Lemma 2.1(i), it is sufficient to show that the cyclic code has minimum distance at most , where . Observe that

Thus, we write

Then

is a codeword of and it has Hamming weight . This implies that has minimum distance at most .

Case 2. .

In this case, we define the cyclic code generated by

Compared with Case , the degree of increases by due to fact that is negative. We can mimic the proof of Case 1 and skip the detail.

Example 3.3.

Let be a prime power. Let be positive integers with , and . We present a -ary optimal cyclic locally repairable code of length , locality and minimum distance with and . The codes in this example were founded in [7] already, but we provide a new view on the construction.

Since , we have . This implies that each irreducible factor of is either linear or quadratic. If an irreducible factor of is quadratic and is a root of , then the other root is . As and , we have . This gives . In conclusion, for each th root of unity , is a polynomial over .

Let be an th primitive root of unity. Let

We first show that is defined over . If is a root of , then either or . This implies that both and are the roots of . Thus, is the product of with some polynomials of the form . We conclude that is defined over . It is clear that since all roots of are th roots of unity and they are distinct. Denote by the cyclic code with generator polynomial . Due to the fact that contains the roots , it follows from Lemma 2.1(i) that has the minimum distance at least . Moreover, the dimension of is . Thus, we have

To show that satisfies the Singleton-type bound (1), it remains to show that the locality of is . By Lemma 3.1, it is sufficient to show that the dual distance of is at most . By Lemma 2.1(i), it is sufficient to show that the cyclic code has minimum distance at most , where . Observe that

Then

is a codeword of and it has Hamming weight . This implies that has minimum distance at most .

3.2 Proof of part (i) of Theorem 1.1

.

Proof.

Let be an th primitive root of unity. Put . Then and , i.e., is an element of . Let .

Note that

This means that is a factor of . Furthermore, since , we have . Let be the cyclic code generated by .

Let us first show that the locality of is . By Lemma 3.1, it is sufficient to show that the Hamming distance of is at most . Put . By Lemma 2.1(i), it is sufficient to show that the Hamming distance of is at most since and are equivalent. Consider the codeword of . As , the Hamming weight of is . Hence, the Hamming distance of is upper bounded by .

Finally, as , are roots of , the minimum distance is at least by Lemma 2.1(iii). By the bound (1), the minimum distance of is upper bounded by . Thus, the minimum distance of is exactly . This completes the proof. ∎

3.3 Proof of part (ii) of Theorem 1.1

.

Proof.

Let be an th primitive root of unity. Put . Then and , i.e., is an element of . Furthermore, is a th primitive root of unity. Since divides , there exist integers such that . Put . Then

As , we must have . Since is a th root of unity, the linear polynomial divides . Furthermore, we have , i.e, is not a factor of . This implies that is a divisor of . Let be the -ary cyclic code of length generated by .

By Lemma 3.1, to show that the locality of is , it is sufficient to show that the Hamming distance of is at most . Put . Then by Lemma 2.1(i), it is sufficient to show that the Hamming distance of is at most since and are equivalent. Consider the codeword of . Since , the Hamming weight of is . Hence, the Hamming distance of is upper bounded by .

Now, we claim that the cyclic code generated by has minimum distance at least . We prove this claim by contradiction. Suppose that the minimum distance of were at most . Then there exists a nonzero polynomial with divisible by . Since are roots of and hence roots of , by Lemma 2.1(ii) we have

(2)

This forces that or or since is an th primitive root of unity. We claim that each of these three divisibilities leads to both and .

Case 1. .

Since are also the roots of , by Lemma 2.1(ii) we have

(3)

By using the condition , Equation (3) is simplified to

Observe that is a primitive th root of unity and . It follows that , or equivalently .

Case 2. .

By using the condition , i.e., , Equation (3) is simplified to

Since , we must have . This gives . Therefore, we have as well.

Case 3. .

In this case, we can mimic the proof of Case 1 by swapping and .

All three cases lead to the conclusion that and . Set and . Then . Moreover, we have the following four identities.

Finally, we notice that are the roots of . By Lemma 2.1(ii), this gives

This is a contradiction due to fact that is a th primitive root of unity and .

The minimum distance of is at most by the bound (1). The proof is completed. ∎

3.4 A locally reparable codes of length and distance

Finally, we present a -ary optimal locally repairable code of length and minimum distance .

Theorem 3.4.

If divides , then there exists a -ary optimal locally repairable code with locality .

Proof.

Let be a th primitive root of unity. Then is an element of . Put . Then and , i.e., is an element of . Let . It is straightforward to verify that is a divisor of . Let be the cyclic code with generator polynomial .

By the similar augments in the proof of Theorem 1.1(i), one can show that dual code of has minimum distance at most .

Since are roots of , has distance at least .

It is easy to check that code meets the Singleton-type bound (1). The proof is completed. ∎

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