Optimal (r, δ)-Locally Repairable Codes From Simplex Code and Cap Code

Locally repairable codes (LRCs) are implemented in distributed storage systems (DSSs) due to their low repair overhead. A linear code <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> is said to have <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula>-locality if for each coordinate <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>, there exists a punctured subcode of <inline-formula> <tex-math notation="LaTeX">$\mathcal {C}$ </tex-math></inline-formula> with support containing <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>, whose length is at most <inline-formula> <tex-math notation="LaTeX">$r+\delta -1$ </tex-math></inline-formula>, and whose minimum distance is at least <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>. An LRC is called optimal if its minimum distance attains Singleton-type bound was proposed. In this letter, optimal LRCs are considered. We first determine <inline-formula> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula>-locality of three dimensional Simplex code, then using anticode strategy, a class of <inline-formula> <tex-math notation="LaTeX">$[{3q,3,2q-1}]_{q}$ </tex-math></inline-formula> LRCs with <inline-formula> <tex-math notation="LaTeX">$(2,q)$ </tex-math></inline-formula> locality are derived for general <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>. Finally, using an ovoid in <inline-formula> <tex-math notation="LaTeX">$PG(3, q)$ </tex-math></inline-formula>, we construct <inline-formula> <tex-math notation="LaTeX">$[q^{2}+1,4,q(q-1)]_{q}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$[{4q-4,4,3q-5}]_{q}$ </tex-math></inline-formula> LRCs with <inline-formula> <tex-math notation="LaTeX">$r=3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\delta =q-1$ </tex-math></inline-formula>. All LRCs constructed in this letter attain the Singleton-type bound.


I. INTRODUCTION
With the increasing demand for data centers on distributed storage systems (DSSs), coding for DSSs have attracted much attention from researchers. Locally repairable codes (LRCs) are a family of erasure codes and designed for DSSs to recover the data in lost node or failed node by accessing r other survived nodes.
LRCs are introduced by Gopalan et al. in [1], some basic properties of LRCs are discussed and a Singleton-type bound on minimum distance of an LRC is proposed in [1]: A code that achieves the bound on the distance with equality will be called an optimal r-local LRC. For optimal r-local LRCs, Tamo and Barg derived a family of optimal LRCs from ''good'' polynomials and data recovery relies on polynomial interpolation [3]. For optimal cyclic r-local LRCs, see Refs.
[4]- [9]. Optimal r-local LRCs over small fields can be seen in [10]- [14]. All the LRCs above can recover data from one failed node at a time, but multiple nodes failures are common in distributed storage system. When some of the r repairing symbols are also erased, the LRC can not finish the repair The associate editor coordinating the review of this manuscript and approving it for publication was Xueqin Jiang . process, which leads to the concept of (r, δ)-locality. Prakash et al. [2] generalized the single node failure solution and presented the (r, δ)-locality to recover data from multiple device failures. A code symbol of a linear code C is said to have (r, δ)-locality if it is contained in a punctured subcode of C, which has length at most r + δ − 1 and minimum distance at least δ. A generalization of Singleton-type bound in [1] is also proposed in [2] LRCs attaining this bound are called optimal with (r, δ)locality and are given in Refs. [15]- [20]. In this letter, two classes of new optimal (r, δ) LRCs are derived from Simplex code with dimension 3 and ovoid code over F q , respectively. Their parameters are [q 2 +q+1, 3, q 2 ] q for r = 2 and δ = q, [3q, 3, 2q − 1] q for r = 2 and δ = q, [q 2 + 1, 4, q(q − 1)] q for r = 3 and δ = q − 1 and [4q − 4, 4, 3q − 5] q for r = 3 and δ = q − 1, respectively. All these LRCs attain the Singleton-type bound.
The rest of this paper is organized as follows: Section II will give the necessary basic concept of LRCs and notations. In section III, the (r, δ)-locality of q-ary Simplex code are determined then with rearrangement of the columns in generator matrix of q-ary Simplex code and the anticode method, optimal [3q, 3, 2q − 1] q LRCs with r = 2 and δ = q are constructed. Section IV deals with an ovoid code and deduce optimal [4q − 4, 4, 3q − 5] q LRCs for r = 3 and δ = q − 1. The last is the conclusion.

II. PRELIMINARIES
In this section, the mathematical notations and definitions are summarized.
Notations 1: 1) If A is a matrix, denote A by the set of columns in A.
2) Let q > 2 be a prime power and F q = {0, x 0 = 1, x 1 , x 2 , · · · , x q−2 } be a finite field with q elements, 3) For any subset X ⊆ F k q , we use X to denote the subspace of F k q spanned by X , where F k q is the k-dimensional vector space over F q . 4) Let w be an element of F * q such that f (y 1 , y 2 ) = y 2 1 + wy 1 y 2 + y 2 2 is irreducible and denote h( is called a linear code and denote by C = [n, k] q . G is the generator matrix of C which can be derived from a basis of C. Let G = (g 1 , · · · , g n ) be generator matrix of C and let G = {g 1 , · · · , g n }. We call the vector in C codeword. The Hamming weight of a codeword c ∈ C is wt(c) = |{i|c(i) = 0}|. The minimum distance of C is defined as d = min Definition 1: [2] The ith code symbol c i , 1 ≤ i ≤ n, in an [n, k, d] linear code C, will be said to have (r, δ)-locality if there exists a subset S i ⊆ [n] such that (1) |S i | ≤ r + δ − 1; (2) The minimum distance of the punctured code C| S i is at least δ.
A famous distance property is introduced in [21] as follows: From Lemma 1, the condition 2 in Definition 1 is equivalent to the following condition Two vectors x and y are equivalent if there exists an non-zero element λ ∈ F q that x = λy. This is an equivalence relation. All the equivalence classes of F k+1 q \{0} form the points of PG(k, q). PG(k, q) is also called k-dimensional projective space.
An n-cap in PG(k, q) is a set of n points no three of which are collinear. If we write the n points of an n-cap in PG(k, q) as columns of a matrix, we obtain a (k +1)×n matrix such that every set of three columns is linearly independent; hence the check matrix of a linear code has minimum distance d ≥ 4. It follows that an n-cap in PG(k, q) is equivalent to a q-ary linear [n, n−k −1, 4] q code. An n-cap in PG(k, q) of maximal size is called a maximal cap in PG(k, q). For more details about caps, see [23]. A code C is called a projective code if the minimum distance d of its dual is equal to or greater than 3.
Next, we introduce the concepts of anti-code and Simplex code.
Definition 3: [22] An anticode A with length n and maximal distance δ is a multiset of codewords in F n q such that the maximum Hamming distance between any pair of codewords is less than or equal to δ. Denote its generator matrix by G, the anticode by A G then we have Definition 4: [23] Denote the matrix S k such that the columns are representatives for the one-dimensional subspaces in F k q . Then we call the code generated by S k q-ary Simplex code with dimension k, denote it by S k . Obviously, the dual of Simplex code is Hamming code.

III. OPTIMAL LRCs WITH r = 2 AND δ = q
This section will discuss the construction of optimal LRCs with parameters r = 2 and δ = q. First, we determine the (r, δ)-locality of q-ary Simplex code with dimension 3, then deduce a class of optimal LRC from three dimensional Simplex code. DIMENSION 3 In order to determine the (r, δ)-locality of q-ary Simplex code with dimension 3, we need to construct a proper generator matrix of Simplex code.
Before constructing LRCs, let us construct the points in PG (2, q) in the form we wish. The first thing to determine is the number of points in PG (2, q) Then G = [I 3 |A 0 · · · A q−2 |B 0 · · · B q−2 ] is the matrix formed by all the points in PG(2, q) and G generates a Simplex [q 2 + q + 1, 3, q 2 ] q code C. The number of columns with weight 1, 2, 3 in C are 3, 3(q − 1), (q − 1) 2 , respectively. In [24], the authors determined the availability of q-ary Simplex code and derived more LRCs using anticode strategy.
Next, we will focus on the parameters δ of q-ary Simplex code with dimension 3, then deduce the (r, δ)-locality of q-ary Simplex code. Further, we deduce a class optimal (r, δ) LRCs, whose length n is field-size. VOLUME 8, 2020 Lemma 2: Let S 3 be q-ary Simplex code with dimension 3, then S 3 have (2, q)-locality and are optimal.
For parameter δ, we use the expression For convenience, denote I 3 = (e 1 , e 2 , e 3 ). From r = 2, let It's not difficult to determine every X i is a 2-dimensional subspace of F 3 q . Considering X 1 , this subspace has (q + 1) vectors as follows From the discussion above, we know the rank of matrix T is 2 and because any two columns in T are distinct, hence any two columns also have the same rank. From the equivalent condition of definition 1, we have constructed where α i and β s are the columns of T and G, respectively. Obviously, |S j | = r + δ − 1 = q + 1. Hence δ = q means any q−1 columns in T can be recovered by left two columns. For the vectors in T , they have (2, q)-locality. With a similar discussion on other columns with weight less than 3, they still have the same locality. For the columns in G with weight 3, we choose a vector with weight 3 in the following form Hence for the columns with weight 3 above, they have (2, q)locality. With a similar discussion for the other columns with weight 3 in G, we obtain the same result. Thus q-ary Simplex code with dimension 3 have (2, q)-locality.
For the optimality of q-ary Simplex code with dimension 3, we need to consider n − k − ( k r − 1)(δ − 1) This subsection will discuss optimal LRCs from q-ary Simplex code with dimension 3 under the anticode strategy. Lemma 3: Let q be prime power, n = 3q, then there exist an optimal [3q, 3, 2q − 1] q LRC with (2, q)-locality.
Let S 3 be q-ary Simplex code with dimension 3 with generator matrix S 3 . Construct there are (q − 1) 2 column vectors in G. In the anticode generated by G, we have Hence, a linear code generated by S 3 \ G have parameters As the discussion of vectors with weight less than 3 in Section III-A, we can prove linear [3q, 3, 2q − 1] q code have (2, q)-locality. The optimality of the obtained code can be derived from the fact that

IV. OPTIMAL LRCS FROM (q 2 + 1)-CAP IN PG(3, q)
In this section, a (q 2 + 1)-cap in PG(3, q) will be constructed first and then the parameters of responding cap code is determined. Finally, using this cap code, a class of optimal LRCs will be obtained. First, a proper construction of (q 2 + 1)-cap is needed.
For any two columns of E 2 , their distance is 2, hence, these two columns and e 3 , e 4 are linear dependent. For the columns in E 3 or J i , any two columns in the same block matrix and e 3 , e 4 are also linear dependent. Hence the locality r of [q 2 + 1, 4, q(q − 1)] q is 3. For parameter δ, we consider another aspect of this ovoid code. The [q 2 +1, 4, q(q−1)] q cap code is a two-weight code with weight q(q−1) and q 2 [26]. Hence for a generator matrix of [q 2 + 1, 4, q(q − 1)] q cap code, there are (q + 1) columns beginning with entry 0. Denote subspace β, e 3 , e 4 of F 4 q by S, where β is a column of E 2 . We can obtain |S ∩ {E 2 , e 3 , e 4 }| = q + 1.
One aspect, any three columns in this q + 1 columns are linear independent from this (q 2 + 1)-cap. Another aspect, these q + 1 columns have rank 3. Hence for code symbols which are located in these (q + 1) columns have parameter With a similar discussion on the columns in block matrix E 3 and J , we can deduce [q 2 + 1, 4, q(q − 1)] q code have (3, δ = q − 1)-locality. Considering we prove the optimality and finish the proof. Next, we present a class of optimal LRCs through modifying [q 2 + 1, 4, q(q − 1)] q code. Lemma 5: Let q be a prime power, there exists an optimal [4q − 4, 4, 3q − 5] q LRC with (3, q − 1)-locality.
Remark 1: In Section III-B and IV-B, we derive two classes of optimal LRCs from anticode. The difference between our anticodes and the ones in [24] is the generator matrices of our anticodes include all the column vectors in G of weight 3 in Lemma III-B and weight 4 in Lemma III-B, while the generator matrices of anticodes in [24] consist of columns weight 2 and more than 2 (Theorem 10, 11 in [24]).

V. CONCLUSION
In this letter, constructions of (r, δ) optimal LRCs are discussed. After analysis of construction of Simplex code, we determined its locality (r, δ) = (2, q) and derived a class of optimal [3q, 3, 2q − 1] LRCs with (2, q). Based on (q 2 + 1)-cap, a [q 2 + 1, 4, q(q − 1)] q LRC with (3, q − 1) was constructed and a class of [4q, 4, 3q−5] q LRC with (3, q−1) was obtained. All the codes attain the Singleton-type bound [2]. The advantage is the length of the obtained codes from Simplex code and cap code is field-size. And for clarifying the difference between our codes and the known results, we make a comparison, see Table 1. Although all the LRCs we constructed are optimal, their code rate are not high enough and code parameters are not flexible. This will lead us to study optimal LRCs with better code rate and flexible parameters in the future.