1 Introduction
The regular generation of vast amounts of data, and the desire to store this data reliably, serve as the impetus for the design of robust distributed storage systems (DSS). Locally repairable codes (LRCs) are a class of codes designed to correct symbol erasures by contacting a small number of other codeword symbols, and have recently attracted a great amount of interest.
We say that an linear code is locally repairable with locality if each codeword symbol is a function of at most other symbols. While small locality is desirable, there is a trade-off between locality and the minimum distance of the code, which we simultaneously seek to keep large in the event of many erasures. It was shown in [1] that an code with locality obeys a Singleton-like bound given by
We call an LRC which meets this bound optimal. The Singleton-like bound naturally calls to mind Maximum Distance Separable (MDS) codes, which meet the Singleton bound of . In particular, an optimal LRC with is an MDS code.
The MDS conjecture states that there are no non-trivial MDS codes with block length larger than , where is the alphabet size of the code, and that in most cases the upper bound is ; the case in which is prime was shown by Ball in [2]. It is thus natural to speculate as to the relationship between alphabet size and block length for LRCs. Early optimal LRC constructions required alphabet size exponential in block length [3, 4]. In [5], Tamo and Barg used subcodes of Reed-Solomon codes to construct optimal LRCs over alphabet size linear in block length; several other constructions also gave block length for LRCs with alphabet size [6, 7, 8].
Barg et al. then presented constructions of length in some cases of small distances [9], distancing the behavior of LRCs from that of MDS codes in this regard. This work was followed closely by the results of [10] which demonstrated optimal LRCs of unbounded length for the cases . Recently, however, Guruswami, Xing, and Yuan [11] showed that for minimum distance at least , the length of an optimal LRC is in fact bounded by a function of the alphabet size; they also gave simpler constructions of unbounded length for .
In this paper, we give two explicit constructions of optimal LRCs with minimum distance that have largest possible asymptotic length as a function of the alphabet size . In this case, the authors of [11] show that the block length is at most , and also showed a greedy construction to achieve it. Concurrent work by Jin gives optimal LRC constructions of length for minimum distances 5 and 6 via binary constant weight codes [12]. The first construction of this paper uses cyclic codes and has similarities to the unbounded-length constructions of [10]. Our second construction uses Cartesian codes.
The basic theory of cyclic codes can be found in, for example, [13]. Loosely speaking, Cartesian codes are obtained when polynomials with variables up to a certain total degree are evaluated on a Cartesian set on components. Each of the components is a subset of the finite field When is Cartesian codes become Reed-Solomon codes. In this paper we will focus on the case when is and each component is the multiplicative group Cartesian codes were introduced, independently, in [14] and [15]. Many properties and applications of Cartesian codes have been studied since their introduction: for example, [16] and [17] investigate Hamming weights and generalized Hamming weights, respectively, and in [18], the authors examine the property of being linear complementary dual.
2 Preliminaries
We first give several definitions and results that will apply to both code constructions. We begin by formally defining locally repairable codes. Throughout the paper, we will focus on linear codes: -dimensional subspaces of with minimum Hamming distance . Let .
Definition 2.1.
Let be a -ary block code of length . For each and , define
For a subset , we denote by the projection of onto . For , a subset of that contains is called a recovery set for if and are disjoint for any , where . Furthermore, is called a locally repairable code (LRC) with locality if, for every , there exists a recovery set for of size .
An optimal LRC with locality is an LRC for which equality is met in the Singleton-like bound given by
By Lemma 2.2 and Remark 2 of [11], achieving equality above is equivalent to the following if :
(1) |
In [11], the authors show the following for the case .
Theorem 2.1 ([11]).
Let be an optimal locally repairable code of locality , with and parameters satisfying
(2) |
Then, .
Theorem 2.2 ([11]).
Assume and . Then there exist optimal LRCs of length . In particular, one obtains the best possible length for optimal LRCs of minimum distance 5.
Note that while these results stipulate that for ease of argument, the authors also explain how to extend to the case in which .
3 Cyclic Code Construction
In this section, we construct an optimal LRC of minimum distance whose length grows quadratically with alphabet size. Indeed, this gives an explicit construction of an LRC with the best possible relationship between alphabet size and block length, per Theorems 2.1 and 2.2 [11]. Our construction is inspired by those of arbitrarily long LRCs for and in [10]. Recall,
Definition 3.1.
An cyclic code is defined as follows by a degree polynomial that divides :
We will use the following properties of cyclic codes:
Theorem 3.1.
Let be a prime power, , , and a primitive th root of unity in . The cyclic code of length generated by
is an optimal LRC with locality and minimum distance .
Proof.
Begin by observing that is indeed a polynomial over that divides . By construction, the alphabet size of this code is , its length is , and its dimension is . With these parameters established, it is clear to see that the code will be optimal provided that its minimum distance is and it has locality , since in this case and
What remains to be shown is that the locality of the code is and its minimum distance is . To show the locality is , it is sufficient to show that the minimum distance of the dual code is at most . Using standard facts about cyclic codes, we have that the weight distribution of the dual code is identical to the cyclic code generated by . Therefore, exhibiting a codeword in of weight at most will be sufficient. Now we simply note that the word given by
has weight .
With locality established, the Singleton-like bound gives an upper bound of on . Since has three consecutive roots of unity as roots (, , and ), we know that the minimum distance of is at least . It is immediate that is a root of , and since
we see that is also a root of .
Suppose there exists some -sparse polynomial that has , , , , and as roots. In other words, letting ,
We claim that has rank
, and thus that the vector
must be the all-zero vector (and so there is no nontrivial word of weight less than or equal to 4). Clearly, the first three rows of are linearly independent due to their Vandermonde structure. Now, to establish rank , it is sufficient to show that one of the fourth or fifth row is independent of the first three.Suppose the first three rows of span the fourth row. We wish to show that if this is the case, the fifth row cannot be spanned by the first three rows. Because the fourth row is some linear combination of the first three, there exists a quadratic such that and for . This implies that the cubic has the property that and for . If the fifth row were spanned by the first three, however, there would exist a quadratic with the above property. Because a cubic cannot match a quadratic on 4 points, we see that the fifth row is not spanned by the first three, completing our proof.
∎
Remark 3.1.
Remark 3.2.
Stipulating that be either large or large compared to in Theorem 3.1 will ensure that the inequality in Equation 2 is satisfied. That is, that the constructed code is of optimal length in addition to being optimal in terms of the trade-off between locality and minimum distance.
Taken with the other assumptions of the theorem, requiring is sufficient to guarantee that the LRC is of optimal length. If or , large and will be necessary to satisfy the inequality.
4 Cartesian Code Construction
In this section we describe an optimal distance LRC over of length using Cartesian codes.
Definition 4.1.
Define and Fix an ordering on the points of Let be the -subspace of spanned by the monomials The Cartesian code on two components is defined by
Observe that the vanishing ideal of is given by . Thus the only element of that vanishes on is the zero element.
For each point of define the polynomial
(3) |
It is straightforward to check that if and only if and if and only if Furthermore, is unique with such a properties, because if there were another polynomial with same properties, then would vanish on but the zero polynomial is the only polynomial in that vanishes on
Lemma 4.1.
Let be an element of Then
where represents the value of at the point
Proof.
Define . Note that for all we have . Therefore vanishes on . However, the only polynomial in with this property is the zero polynomial, which implies ∎
The following result is key to finding LRCs using Cartesian codes.
Lemma 4.2.
If , are distinct elements of then the matrix
has linearly independent columns.
Proof.
If then the ’s are distinct. Thus the first four rows are multiples of the rows of a Vandermonde matrix with distinct elements, which means the columns of are linearly independent. If all the ’s are pairwise distinct, divide each column by the element Thus rows and form a Vandermonde matrix with distinct elements, so again the columns of are linearly independent.
If then and are distinct. Dividing each row by the power of that appears in its first entry, and using row to simplify rows and , becomes the matrix
where Then row is non-zero and all the columns are linearly independent, because columns and contain a Vandermonde matrix with distinct elements, and the first three columns cannot span the fourth.
If and then and are different. For , divide column of by and subtract the second column from the first. Matrix becomes the following:
As position is non-zero. Thus, the columns of are linearly independent because columns and contain a Vandermonde matrix with distinct elements, and these three columns cannot span the first.
Finally, if then and Subtract the first column from the second one, the third column from the fourth, and then the first column from the third. Divide each row by the power of that appears in its first entry; divide the second column by and the fourth column by where Then becomes:
Subtracting row from rows and we obtain:
It is clear that columns 1 and 2 are independent. As position is non-zero, thus columns and are linearly independent. If columns 1, 2, and 4 span column 3, then column must be a multiple of column Since , rows 5, 6, and 7 imply this multiple must be . However, this would imply that in the second row, a contradiction. Thus, all columns are independent. ∎
Theorem 4.3.
Assume . Let be the -subspace of spanned by the monomials
The code is an LRC with locality and minimum distance If , is an optimal LRC with locality and minimum distance
Proof.
Consider the sets of the form , where . These partition into disjoint sets, each of size ; we claim these are recovery sets of the code. Evaluating all monomials of at reduces to polynomials in of degree less than Therefore there is a single parity check equation for the points in each set , and so these form recovery sets.
Now we prove that the minimum distance of is at least Let be an element of If has only non-zero evaluations on by Lemma 4.1 there are and for such that
By (3), the coefficient of the monomial in is given by As the monomial does not belong to the support of then As does not belong to the support of and then In a similar way, as the monomials , , , , do not belong to the support of , and its coefficients are of the form then the following matrix has linearly dependent columns
This is not possible, by Lemma 4.2. Using same matrix we conclude cannot have only non-zero elements in , because in such a case, matrix would have linearly dependent columns. In addition, cannot have only non-zero elements in , because in such a case, matrix would have linearly dependent columns. Finally cannot have only one non-zero element in because by (3), its support should contain all the monomials. Thus, the minimum distance of the code is at least 5.
Recall that Thus, when , by the Singleton-like bound the minimum distance of is at most . Since we have shown that the minimum distance is at least , is an optimal LRC when . ∎
5 Conclusions
In this paper, we used cyclic and Cartesian codes to construct two families of optimal-length, optimal LRCs for the case in which the minimum distance is equal to 5. Ongoing work includes extending our arguments to higher minimum distance and exhibiting other algebraic constructions of optimal-length LRCs.
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