Optimal Linear and Cyclic Locally Repairable Codes Over Small Fields

Optimal Linear and Cyclic Locally Repairable Codes over Small Fields

Alexander Zeh and Eitan Yaakobi Computer Science Department Technion—Israel Institute of Technology [email protected], [email protected]

Abstract—We consider locally repairable codes over small This paper is structured as follows. Section II gives fields and propose constructions of optimal cyclic and linear codes necessary preliminaries on linear and cyclic codes, defines in terms of the dimension for a given distance and length. LRCs, recalls the generalized Singleton bound, the Cadambe– Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two Mazumdar bound [8] as well as the definition of availability approaches give binary cyclic codes with locality two. While the for LRCs. The concept of a locality code and the projection first construction has availability one, the second binary code is to an additive code are discussed in Section III based on the characterized by multiple available repair sets based on a binary work of Goparaju and Calderbank [9]. Two new constructions Simplex code. of optimal binary codes are given in Section IV and a con- The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property struction based on a q-ary shortened cyclic first-order Reed– is determined by the properties of a shortened first-order Reed– Muller code is given in Section V. The fourth construction Muller code. Non-cyclic optimal binary linear codes with locality in Section VI uses code concatenation and provides optimal greater than two are obtained by the fourth construction. linear binary codes. Section VII concludes this contribution. Index Terms—Availability, distributed storage, locally repairable codes, Reed–Muller code, Simplex code, sphere-packing II.PRELIMINARIES bound Let [a, b) denote the set of integers a, a + 1, . . . , b 1 { − } I.INTRODUCTION and [b) be the shorthand notation for [0, b). Let Fq denote the Locally repairable codes (LRC) can recover from erasure(s) finite field of order q and Fq[X] the polynomial ring over Fq by accessing a small number of erasure-free code symbols and with indeterminate X. A linear [n, k, d]q code of length n, therefore increase the efficiency of the repair-process in large- dimension k and minimum Hamming distance d over Fq is denoted by a calligraphic letter like as well as a non-linear scale distributed storage systems. Basic properties and bounds C of LRCs were identified by Gopalan et al. [1], Oggier and (n, M, d)q of length n, cardinality M and minimum distance Datta [2] and Papailiopoulos and Dimakis [3]. The majority d. of the constructions of LRC requires a large field size (see An [n, k, d]q q-ary cyclic code with distance d is an ideal n C in the ring Fq[X]/(X 1) generated by g(X). The generator e.g. [4]–[6]). The work of Kuijper and Napp [7] considers − binary LRCs (and over binary extension field). Cadambe and polynomial g(X) has roots in the splitting field Fqs , where n (qs 1). Mazumdar [8] gave an upper bound on the dimension of | − a (nonlinear) code with locality which takes the field size A q-cyclotomic coset Mi,n is defined as into account. Goparaju and Calderbank [9] proposed binary def M = iqj mod n j [a) , (1) cyclic LRCs with optimal dimension (among linear codes) for i,n | ∈ arXiv:1502.06809v1 [cs.IT] 24 Feb 2015 distances 6 and 10 and locality 2. where a is the smallest positive integer such that iqa i mod Our paper is based on the work of Goparaju and Calder- ≡i n. The minimal polynomial in Fq[X] of the element α Fqm bank [9] and we use their projection to an additive code Q j ∈ is given by mi(X) = (X α ). The defining set DC without locality (see Calderbank et al. [10], Gaborit et al. [11], j∈Mi,n − of an [n, k, d] cyclic code is Kim et al. [12] for additive codes). We construct a new family q C i of optimal binary codes (with distance 10 and locality 2) DC = 0 i n 1 g(α ) = 0 . (2) and generalize the approach to q-ary alphabets. Furthermore, ≤ ≤ − | we give a construction of optimal binary cyclic codes with For visibility we sometimes mark a position i with a if [z] availability greater than one based on Simplex codes (see g(αi) = 0. Furthermore, let D be the short-hand notation 6 C Pamies-Juarez et al. [13], Rawat et al. [14] for the definition for (i + z) i DC for a given z Z. Let us recall the { | ∈ } ∈ of availability and Kuijper and Napp [7] for a Simplex code definition of linear locally repairable codes. based construction). Definition 1 (Locally Repairable Code (LRC)). A linear [n, k, d] code is said to have (r, δ)-locality if for all n This work has been supported by German Research Council (Deutsche q C Forschungsgemeinschaft, DFG) under grant ZE1016/1-1. code symbols c , i [n), there exists a punctured subcode of i ∀ ∈ with support containing i, whose length is at most r +δ 1, but does not increase the overall erasure-correction capability. C − and whose minimum distance is at least δ. In general, let be the aimed [n, k, d]q code with locality properties that areC inherited from an [n , k , d ] locality code A code is called r-local if it has (r, 2)-locality. l l l q . Namely the code constructions of [9] and our (cyclic) codes The followingC generalization of the Singleton bound for areL subcodes of the [n, k, d] cyclic product code , where LRCs was among others proven in [15, Thm. 3.1], [6, Con- q is the trivial [n/n , n/n , 1] code (see [17]),L⊗T i.e., a cyclic struction 8 and Thm. 5.4] and [16, Thm. 2]. l l q codeT with defining set: The minimum Theorem 2 (Generalized Singleton Bound). n [nl] [n−nl−1] o DC = DL DL DL R . (5) distance d of an [n, k, d]q linear (r, δ)-locally repairable code ∪ ∪ · · · ∪ ∪ (as in Def. 1) is upper bounded by In Construction 2 (R = M1 ) and 3 (R = M1 M−1 ) of C ,n ,n ∪ ,n k [9], the locality code is a [3, 2, 2]2 single-parity check code d n k + 1 1 (δ 1). (3) L ≤ − − r − − with defining set DL = 0 . The optimality among binary codes with locality r = 2{is} shown via the projection to an For δ = 2 and r = k it coincides with the classical Singleton additive code. bound. Throughout this contribution we call a code Singleton- optimal if its distance meets the bound in Thm. 2 with equality. Lemma 5 (Projection to Additive Code). Let be an [n , k , d ] locality code and let be an [n, k, d] codeL with The generalized Singleton bound as in Thm. 2 does not take l l l q C q the field size into account. We compare our constructions with defining set as in (5). Then, we can project each sub-block of the bound given by Cadambe and Mazumdar [8, Thm. 1] nl symbols of a codeword in to one symbol in Fqkl . The 0 k 0 C which depends on the alphabet size. In general it holds also obtained (n , q , d ) kl additive code has parameters: q A for nonlinear codes, but we state it only for linear codes in 0 0 n = n/nl and d d/ω , (6) the following. ≥ d e where ω is the maximum weight of a codeword in . Theorem 3 (Cadambe–Mazumdar (CM Bound)). The dimen- L 0 sion k of an r-local repairable code of length n and Proof: The length n and the alphabet-size follow directly minimum Hamming distance d is upper boundedC by from the projection of the coordinates. The cardinality of equals the one of . The distance follows from the fact thatA in n (q) o C k min tr + kopt n t(r + 1), d , (4) the worst-case ω non-zero symbols of are projected to one ≤ t∈Z − C symbol over Fqkl (see Fig. 1). (q) where kopt(n, d) is the largest possible dimension of a code of length n, for a given alphabet size q and a given minimum n 1 /nl distance d. In the following we use Thm. 3 to bound the dimension 1 ? ··· of linear codes. Another important parameter for LRCs is ...... the availability e.g. considered in Kuijper and Napp [7] and ...... Cadambe and Mazumdar [8] and therefore we define it in the k ? ? following. l ··· Definition 4 (Availability). An [n, k, d]q linear code is ? ? called t-available-r-local locally repairable if every codeC ··· ...... symbol ci, i [n), has at least t parity-checks of weight ...... r + 1 which∀ intersect∈ pairwise in (and only in) i . { } nl ? ? III.LOCALITY CODEAND ADDITIVE CODE ··· ∈ ∈ In this section we shortly recall the approach of Goparaju and Calderbank [9] and extend it to what we call a locality code. L ··· L Let us first describe the idea of Constructions 1 and 2 of [9] in terms of a locality code. Construction 1 of [9] gives a Fig. 1. Illustration of a nonzero minimum-weight codeword of weight seven binary cyclic code of length n = 2m 1 with locality r, of the [n, k, d]q code C arranged in n/nl blocks of length nl. The ? marks a nonzero symbol in . The corresponding codeword of the additive code A where the code lengthC n is divisible by r +1− . The defining set Fq over Fqkl has length n/nl and has at least weight dd/ωe. Here ω = 4 and is DC = i mod (r + 1), i [n) . This equals the union of therefore at least two symbols in A are nonzero (first and before last column). { ∀ ∈ } The redundancy added by the [n , k , d ] locality code L is illustrated as gray n/(r + 1) shifted defining sets DL = 0 of the binary cyclic l l l q { } symbols, while the black symbol marks the additional redundancy to obtain [r + 1, r, 2]2 single-parity check code, which is able to correct a distance that is higher than the one given by the Singleton bound. one erasure within a block of length r +1 by “accessing” only r other code symbols. The code inherits the properties of , For a cyclic binary [r + 1, r, 2]2 single-parity check code of also the minimum distance of two,C which is Singleton-optimal,L odd length, the maximum weight of a codeword is ω = r. m Lemma 6 (Locality Code). If is an [nl = r + δ 1, r, δ]q With n = 2 + 1, we obtain from (9) MDS locality code, then the cyclicL code with defining− set as C 2n m m 2 in (5) has (r, δ)-locality and its distance is d δ. k = + 1 log2 2 (2 + 1) + (2 + 1) ≥ 3 − d − e Proof: The (r, δ)-locality of follows directly from the 2n m 2m = + 1 log2 2 + 2 + 2 construction. The distance of C is at least the distance 3 − d e C ⊆ L⊗T 2n 2n of the product code . = + 1 (2m + 1) = 2m. L ⊗ T Note that for R = , the code = is Singleton- 3 − 3 − optimal, i.e., the dimension∅ of isCk =Lnr/ ⊗( Tr + δ 1) and from (3) we obtain: C − Remark 1: A binary cyclic code as in Construction 7 without M1,n in the defining set is Singleton-optimal and the nr n distance equals d = 2 (for k = 2n/3, r = 2 and δ = 2), which d n + 1 1 (δ 1) ≤ − r + δ 1 − r + δ 1 − − is the smallest minimum distance possible for a binary cyclic − − n(r + δ 1) nr n(δ 1) code with rate 2/3 (see [18, Prop. 2]). − − − − + δ = δ. ≤ r + δ 1 Remark 2: The nth root of unity is in F22m−1 (which is twice − the extension order of the code obtained via [9, Construction IV. BINARY CYCLIC CODES WITH LOCALITY TWO 3]) and therefore the (non-local) decoding complexity is higher than [9, Construction 3]. Construction 1 in [9] gives Singleton-optimal binary cyclic 5 codes (these codes are of lowest-rate for d = 2 see [18, Prop. Example 9 (Optimal 2-Local Binary Code). Let n = 2 +1 = 2]). The following construction gives a new class of binary 33 and via Construction 7 we obtain a binary cyclic code of cyclic 2-local codes. minimum distance d = 10, with Construction 7 (Binary Reversible Codes). Let n = 2m + 1 DC = 0, 3, 6, 9,..., 30 1, 2, 4,..., 32 { } ∪ { } and 3 n and therefore m odd. Let the locality r = 2, i.e. let | and dimension k = 12, which is 2-local. The CM bound (see be a [3, 2, 2]2 single-parity check code with DL = 0 . Let L { } Thm. 3) based on the best-known linear codes give k 13. the defining set be ≤ Let us consider Construction 4 of [9] based on the [7, 3, 4] 2 DC = ..., 6, 3, 0, 3, 6, 9, 12,... M1,n , locality code with locality r = 2, availability t = 3 and { − − } ∪ L = ..., 6, 4, 3, 2, 1, 0, 1, 2, 3, 4, 6,... . with defining set DL = 0, , , 3, , 5, 6 . The code with { − − − − − } defining set as in (5), but with R = is an [n = 2m C1, k = 2 m ∅ − Then has dimension k = 3 (2 + 1) 2m and distance 3n/7, 4]2 cyclic code, where m is a multiple of three. d 10C. − We extend Construction 4 of [9] to obtain a higher distance ≥ and small reduction of the rate as follows. Due to the length, the coset M1,n is reversible (see [19] m−1 m for reversible codes), i.e., M1,n = 1, 2,..., 2 , 2 = Construction 10 (Sphere-Packing Optimal Binary Code with 1, 2,..., 2m−1 and has cardinality{ 2m. The distance Locality Two and Increased Availability). Let n = 2m 1 and − follows− − from− the BCH} bound [20], [21], where the consecutive be divisible by 7 and therefore 3 m. Let the defining set be: | sequence is 4, 3,..., 3, 4. − − DC = ..., 9, 8, 7, , , 4, , 2, 1, 0, , , { − − | − − − − | Let a binary linear code with parameters as in Theorem 8. 3, , 5, 6, 7, , , 10, , , 12, 13 ,... Construction 7 be given. Then its dimension satisfies: | m−1 | } M1 = 1, 2, 4,..., 2 . ∪ ,n { } 2 k (2m + 1) 2m. (7) Then d 12 (via BCH bound [20], [21], where the con- ≤ 3 − secutive≥ sequence is 2, 1,..., 8). The constructed [n = m − 2 1, k = 3n/7 m, d 12]2 cyclic code is a 2-local Proof: Equivalent to the proof of [9, Thm. 2], we − − ≥ C have via sphere-packing bound (see [22, Ch. 1 5]) for the code and has availability t = 3 as defined in Def. 4. k § (2n/3, 2 , 5)22 additive code that Example 11 (2-Local Binary Code with Availability Three). 0 Let n = 9 7 = 63, DL = 0, , , 3, , 5, 6 and let the 4n · 2k (8) defining set be ≤ 0 9n0(n0−1) 1 + 3n + 2 [7] [56] DC = DL D D M1 63 { ∪ L ∪ · · · ∪ L ∪ , } and therefore = .., 59, 61, 62, 0, 3, 5, 6, 7, 10, .. 1, 2, 4, .., 32 . {{ | | } ∪ { }} 1 1 n/3 2 The constructed code is an [63, 21, 12]2 code and the k log2(4 ) log2 1 n + n ≤ − − 2 2 corresponding additiveC code according to Lemma 5 is a 7 2n 2 (9, 2 , 3) 3 code. The BCH bound is tight and the consecutive = + 1 log 2 n + n . (9) 2 3 − d 2 − e sequence ranges is 61, 62, 0, 1,..., 8. Let us prove the optimality of Construction 10 in the V. Q-ARY CASE:FIRST-ORDER SHORTENED RMCODEAS following theorem. LOCALITY CODE m Theorem 12. Let 3 m and let be an [2 1, k, 12]2 linear We extend the previous approach for cyclic codes to the | C − code, let be the [7, 3, 4]2 Simplex code and let have the q-ary case and use as locality code the q-ary L C L locality inherited from as in Lemma 5. Then, 2 2−1 2 2 L [q 1, 2, (q 1)q ]q = [q 1, 2, q q]q (13) 3 − − − − k (2m 1) m. (10) ≤ 7 − − cyclic shortened first-order Reed–Muller (RM, see [24, Prob- 0 k 0 lem 2.17] and [25, Section 6.11]) code. Its dual code is Proof: From (6) we have an (n = n/7, 2 , d = 2 2 the [q 1, q 3, 2]q code with defining set 1, q .A 12/4 = 3) 3 additive code . The [7, 3, 4] Simplex code 2 2 a − a−1 − shortened first-order RM{ code} is dis a constant-weighte code and thereforeA ω = 4. The code is [q 1, a, q (q 1)]q A the −q-ary pendant of− the Simplex code and also a constant- defined over F23 and has dimension weight code with ω = qa−1(q 1). RM codes have the highest 1 3 − k0 = k/3 = n m = n0 m0. minimum distance possible for the given parameters among q- 3 7 − − ary linear codes. Furthermore, first-order RM codes and their 0 and therefore the parameters of a [((23)m 1)/(23 1), n0 locality properties were investigated by Rawat and Vishwanath 0 − − − m , 3]23 Hamming code, which is optimal w.r.t. the sphere- in [26]. packing bound. Construction 15 (Reed–Muller Code Locality). Assume q > Construction 10 can be extended to the case where the 2. Let be an [qm 1, k, d] code. Let be an [q2 1, 2, q2 locality code is the a a−1 cyclic Simplex code, q [2 1, a, 2 ]2 q] cyclicC RM code− with defining set L − − which is 2-localL and has− availability t = 2a−1 1 (see e.g. q − 2 Kuijper–Napp [7, Lemma 3.1] and Wang–Zhang [23]). DL = 0, , 2, 3, . . . , q 1, , q + 1, . . . , q 2 . − − Construction 13 (Binary Code with Simplex Locality). Let Let the defining set of be: n = 2m 1 and be divisible by 2a 1 and therefore a m. Let C n [q−1] 2 m−1 o − a a−1 − | DC = DL D 1, q, q , . . . , q be the [2 1, a, 2 ]2 cyclic Simplex code with defining ∪ L · · · ∪ { } Lset − 2 2 = , (q q 2), .., 0, .., q + q 2, , .. . a−1 a { − − − } DL = 0, , , 3, , 5, 6, .., , 2 + 1, .., 2 1 . (11) 2n Then the dimension of is k = 2 m and the distance − C q −1 − Let the defining set of the code be: is d q2 q 2 + q2 + q 2 + 1 + 1 = 2q2 2 via BCH C ≥ − − 2− 2 − n [2a−1] [2(2a−1)] o bound [20], [21] (from (q q 2).. + (q + q 2)). DC = DL D D M1,n − − − − ∪ L ∪ L ∪ · · · ∪ a−1 a a Theorem 16 (Locality Code: Shortened First-Order RM = ..., 2 + 1,..., 1 , 0, 1,..., 2 1, 2 ,... . m 2 − − | − | Code). Let be an [q 1, k, (q q) 3]2 linear code, Then d 2a + 2a−1 (via BCH bound for the consecutive let be theC[q2 1, 2, q2 − q] RM code− and· let have the ≥ q sequence from (2a−1 1) to 2a) and the dimension is k = localityL properties− according− to as in Lemma 5.C Then, a (2m 1) −m. − L 2a−1 − − 2(qm 1) k − m. We have the following theorem on the optimality of the ≤ q2 1 − dimension of linear codes. − Proof: The additive code has parameters A Theorem 14 (Simplex Locality). Let a m and let be an m 0 q 1 m a−1 linear code, let be| the a C a−1 n = − , [2 1, k, 2 3]2 [2 1, a, 2 ]2 q2 1 binary− Simplex· code and let haveL the locality− properties − k0 = k/2, according to as in Lemma 5.C Then, L a 0 d k (2m 1) m. (12) d = , a q2 q ≤ 2 1 − − − − a Proof: From (6) we have an (n/(2 and has alphabet-size Fq2 . More explicitly, the dimension is: k a−1 a − 1), 2 , d/(2 ) )2a additive code , where ω = 2 1, d e A − 1 2(qm 1) because the simplex code is a constant-weight code. 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