Table Of ContentOptimal Binary Locally Repairable
Codes via Anticodes
Natalia Silberstein and Alexander Zeh
Computer Science Department
Technion—Israel Institute of Technology
Haifa 32000, Israel
natalys@cs.technion.ac.il, alex@codingtheory.eu
Abstract—This paper presents a construction for several a new bound for LRCs which takes the size of the alphabet
familiesofoptimalbinarylocallyrepairablecodes(LRCs)with intoaccountwasestablishedbyCadambeandMazumdar[1,
small locality (2 and 3). This construction is based on various
Thm. 1]. They showed that the dimension k of an [n,k,d]
anticodes.ItprovidesbinaryLRCswhichattaintheCadambe–
LRC C over F with locality r is upper bounded by
Mazumdar bound. Moreover, most of these codes are optimal q
5 with respect to the Griesmer bound. k ≤ min(cid:110)tr+k(q)(cid:0)n−t(r+1),d(cid:1)(cid:111), (2)
1 I. INTRODUCTION t∈Z+ opt
0
2 Locally repairable codes (LRCs) are a family of erasure where k(q)(n,d) is the largest possible dimension of a code
opt
n codes which allow local correction of erasures, where any oflengthn,foragivenalphabetsizeq andagivenminimum
code symbol can be recovered by using a small (fixed) num-
a distance d. This bound applies to both linear and nonlinear
J ber of other code symbols. The concept of LRCs was mo- codes. In the case of a nonlinear code, the parameter k is
8 tivated by application to distributed storage systems (DSSs), defined as |C|/logq. Moreover, it was shown in [1] that
2 (see e.g. [2], [3] and references therein). DSSs store data
the family of binary simplex codes attains the bound (2) for
across a network of nodes in a redundant form to ensure
r = 2. To the best of our knowledge, there are only three
]
resilience against node failures. Using of LRCs to store data
T additionalworksthatconsiderconstructionsofbinaryLRCs,
in DSSs enables to repair a failed node locally, i.e., by
I namely the papers of Shahabinejad et al. [19], Goparaju
. accessing a small number of other nodes in the system.
s and Calderbank [7], and Zeh and Yaakobi [22], where the
c The ith code symbol c , 1 ≤ i ≤ n, of an [n,k,d] linear
i constructions of [7] and [22] consider binary cyclic LRCs.
[
code C is said to have locality r if c can be recovered by
i In this paper we propose constructions of new binary
1 accessing at most r other code symbols. A code C is said
LRCs which attain the bound (2). All our LRCs have a
v to have locality r if all its symbols have locality r. Such
small locality (r = 2 and r = 3), moreover, most of our
4 codes are referred to as locally repairable (or recoverable)
1 codes attain the Griesmer bound. Our constructions use a
codes (LRCs). LRCs were introduced by Gopalan et al.
1 method of Farrell [4] based on anticodes. In particular, we
in [6]. It was shown in [6] that the minimum distance of
7 modify a binary simplex code by deleting certain columns
0 an [n,k,d] LRC with locality r should satisfy the following
from its generator matrix. These deleted columns form an
. generalization of the Singleton bound
1 anticode. We investigate the properties of anticodes which
0 (cid:24)k(cid:25) allow constructions of LRCs with small locality. Also, we
5 d≤n−k+2− . (1)
r present optimal binary LRCs with locality 2 based on sub-
1
space codes.
: Constructions of LRCs which attain this bound were pre-
v Therestofthepaperisorganizedasfollows.InSectionII
sentedin[5],[6],[20],[21].FurthergeneralizationsofLRCs
i
X tothecodeswhichcanlocallycorrectmorethanoneerasure, we provide the necessary definitions, in particular, we define
anticodes and describe a method of constructing a new code
r to the LRCs with multiple repair alternatives, and to the
a based on a simplex code and an anticode. In Section III
vector LRCs were considered in [8], [9], [11]–[16], [18],
we present our constructions based on various choices of
[20].
However,toattainthebound(1)anditsgeneralizations[8], anticodes. Conclusion is given in Section IV.
[12], [13], [16], the known codes should be defined over a
II. PRELIMINARIES
large finite field. In [20] Tamo and Barg presented LRCs
satisfying the bound (1), which are defined over a field of Let C be a linear [n,k,d] code of length n, dimension k
size slightly greater than the length of the code. This is the and minimum Hamming distance d over F . We say that a
q
smallest field size for optimal LRCs known so far. k ×n generator matrix G of C is in a standard form if it
Codes over small (especially binary) alphabets are of a has the form G =(I |A), where I is an identity matrix of
k j
particularinterestduetotheirimplementationease.Recently, order j and A is a k×(n−k) matrix. If G is in a standard
formthenan(n−k)×nparity-checkmatrixH canbeeasily
N. Silberstein has been supported in part at the Technion by a Fine obtained from G in the following way: H = (−AT|I ).
Fellowship. A. Zeh has been supported by the German research council n−k
(DeutscheForschungsgemeinschaft,DFG)undergrantZe1016/1-1. Notethatforabinarycode,wehaveH =(AT|In−k).Inthe
sequel, we will consider only binary codes. The following Construction 1 (Farrell Construction [4]). Let G be the
m
simple lemma shows a sufficient condition on a parity-check m×(2m−1) generator matrix of a binary simplex code
matrix of a code with locality r. S and let G be the k×n generator matrix with distinct
m A
columns of a binary linear anticode A of length n and
Lemma 1. An[n,k,d]linearcodehaslocalityr ifforevery
maximum distance δ. Then, the m×(2m −1−n) matrix
coordinate i, 1 ≤ i ≤ n, there exists a row R of weight at
i obtained by deleting the n columns of G from G is a
most r+1 in its parity-check matrix, which has a nonzero A m
generator matrix of a binary [2m−1−n,≤ m,2m−1−δ]
entry in the ith coordinate. In this case we say that the
code.
coordinate i is covered by the row R .
i
Example 2. Let G be the 4 × 15 generator ma-
The following two bounds will be used in the sequel. The 4
trix of a simplex code S and let G be the gen-
Plotkin bound (Thm. 1) holds for nonlinear codes, while the 4 A
erator matrix of the anticode given in Example 1
Griesmer bound is restricted to linear codes (Thm. 2).
with the additional first row of zeros. By deleting the
Theorem1(PlotkinBound[10,p.43]). LetA (n,d)denote columns of G from G we obtain the following matrix
2 A 4
the largest number of codewords in a binary code of length
1 0 0 0 1 1 1 0 0 0 1 1 1 0 1
n and minimum distance d. If d is even and 2d>n then
0 1 0 0 1 0 0 1 1 0 1 1 0 1 1
(cid:22) d (cid:23) G4\GA= 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1
A (n,d)≤2 .
2 2d−n 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1
Theorem2(GriesmerBound[10,p.547]). Thelengthnofa wheretheshadowedcolumnsin{8,9,10}aredeleted,which
binarylinearcodewithdimensionkandminimumdistanced generates a [12,4,6] code. Note that this code attains the
must satisfy Griesmer bound.
k−1(cid:24) (cid:25)
(cid:88) d
n≥ . III. CONSTRUCTIONSOFOPTIMALBINARYLRCS
2i
i=0 In this section we provide constructions of binary LRCs
Intheremainingpartofthissectionwedefineananticode based on the Farrell construction (see Construction 1), by
and recall the anticode-based construction of binary linear usingvariousanticodes.Weprovethatourcodeshaveasmall
codes by Farrell [4] (see also [10, p. 548]). An anticode is locality(r =2orr =3)andattaintheCadambe–Mazumdar
a code which may contain repeated codewords and which bound(2).MostofourcodesalsoattaintheGriesmerbound
has an upper bound on the distance between the codewords. (see Thm. 2).
More precisely, a binary linear anticode A of length n and
A. LRCs based on Anticodes
maximum distance δ is a set of codewords in Fn such that
2
First, we generalize Example 1 and consider an anticode
theHammingdistancebetweenanypairofcodewordsisless
than or equal to δ. The generator matrix G of A is a k×n such that all the vectors of length s and weight 2 form the
A
binary matrix such that all the 2k combinations of its rows columns of its generator matrix. We denote such an anticode
form the codewords of the anticode. If rank(G ) = γ, then by As,2. For example, the anticode from Example 1 is an
A
each codeword occurs 2k−γ times in the anticode. Due to A3,2 anticode. First, we need the following theorem about
the parameters of A .
linearity, we have s,2
Theorem 3. Let A be a binary anticode such that all
δ =maxwt(a), (3) s,2
a∈A weight-2vectorsoflengthsformthecolumnsofitsgenerator
matrix G . Then A has length (cid:0)s(cid:1) and maximum weight
where wt(v) denotes the Hamming weight of a vector v. (cid:106) (cid:107)A s,2 2
δ = s2 .
Example 1. Let GA be a 3×3 generator matrix given by 4
1 1 0 Proof: There are (cid:0)s(cid:1) vectors of weight 2 and length s,
2
GA=10 01 11. h(cid:0)se(cid:1)n.cNetehxet,nwume pbreorvoeftchoaltutmhensvianluGeAofatnhdetmheaxleinmguthmowfeAigs,h2tiδs
It generates a binary linear anticode A of length n=3 and is2(cid:106)s2(cid:107). Note that the s×(cid:0)s(cid:1) generator matrix G is also
δ =2, where the set of 23 codewords is 4 2 A
an incidence matrix of a complete graph K = (V,E) with
s
A=(cid:8)(000),(110),(101),(011),(011),(101),(110),(000)(cid:9). |V| = s and |E|=(cid:0)s(cid:1). Therefore, the maximum weight δ
2
of the anticode can be described in terms of maximum cut
The construction of Farrell [4] which we use to construct between a subset of vertices S ⊆V and its complement Sc,
optimal LRCs is based on a modification of a generator
more precisely,
matrix for a binary simplex code.
AbinarysimplexcodeSmisa[2m−1,m,2m−1]codewith δ =1m≤ia≤xs|Cut(Si,Sic)|,
generator matrix G whose columns consist of all distinct
m
nonzero vectors in Fm. In the rest of the paper we assume whereSi isasubsetofV ofsizeiandSic itscomplementof
2
size s−i. Since K is an (s−1)-regular graph, it holds that
w.l.o.g.thatG isinthestandardformandthatthecolumns s
m
for all 1 ≤ i ≤ s, (s−1)i = |Cut(S ,Sc)|+2|E |, where
of Gm are ordered according to their Hamming weight. i i i
E ⊆E is the set of edges between the vertices in S . Note
i i
thattheinducedsubgraph(S ,E )ofK isacompletegraph we obtain m rows of weight 3 in H(cid:48) with the first nonzero
i i s
K and then |E |=(cid:0)i(cid:1). Thus, entry in the first m coordinates and the last nonzero entry in
i i 2
the last coordinate. Thus, the obtained parity-check matrix
|Cut(S ,Sc)|=(s−1)i−i(i−1)=i(s−i)
i i H(cid:48) contains weight-3 rows which cover all the coordinates
and (and the last row of H, of weight m+1).
(cid:22)s2(cid:23)
δ = max{i(s−i)}= .
1≤i≤s 4 In the following example we illustrate the idea of mod-
ification of a parity-check matrix described in the proof of
As a consequence of Thm. 3 and the Farrell construction Thm. 4.
we have the following theorem. Example 3. We consider the parity-check matrix H of the
Theorem 4. Let S be a [2m−1,m,2m−1] simplex code, [12,4,6] code of Example 2 based on the anticode A3,2. It
m
has the following form, where the vertical lines show the
m≥4,andletG beitsgeneratormatrix.LetA ,s≤m,
m s,2
partition of its columns into the parts H1,H2,H3,H4.
beananticodedefinedinThm.3andletG beitsgenerator
A
matrix. We prepend m−s zeros to every column of GA to 1100 100 0000 0
formanm×(cid:0)s(cid:1)matrixwhosecolumnsaredeletedfromG 1010 010 0000 0
2 m 1001 001 0000 0
to obtain a generator matrix G for a new code C . Then
C isa[2m−(cid:0)s(cid:1)−1,m,2m−1−(cid:106)s2(cid:107)]LRCwmit,hs,2locality H= 11111001 000000 10010000 00
rm=,s2,2. 2 4 10011111 000000 00001001 00
1111 000 0000 1
Proof: The prepending of zeros to G does not change
A
We add to each one of the ith rows of H, 4 ≤ i ≤ 7, the
the length and the maximum weight of the anticode. Hence,
last row and obtain the following parity-check matrix:
the length, the dimension, and the minimum distance of
the obtained code Cm,s,2 directly follow from the Farrell 1100 100 0000 0
construction and Thm. 3. 1010 010 0000 0
1001 001 0000 0
To prove that the locality of C is r =2, by Lemma 1
we need to show that every coormd,isn,2ate is covered by a row H(cid:48)= 00000110 000000 10010000 11 ,
of weight 3 of the parity-check matrix for C (note that 0100 000 0010 1
m,s,2 1000 000 0001 1
clearly locality is not 1). Since G is in the standard form,
m 1111 000 0000 1
thegeneratormatrixGofC isalsointhestandardform.
m,s,2 such that every coordinate is covered by a weight-3 row.
We denote by Gi, 1 ≤ i ≤ m, the submatrix of G which
consists of the set of columns of G of weight i. Then the Corollary 1. For 3 ≤ s ≤ 5, the code Cm,s,2 obtained in
parity-check matrix H of C has the following form: Thm. 4 attains the bound (2). More precisely,
m,s,2
(G2)T I • The code Cm,3,2 obtained by using the anticode A3,2 is
(m)−(s) a [2m−4,m,2m−1−2] LRC with locality r =2 which
2 2
H = ... ... . attains the bound (2).
(Gm−1)T I • The code Cm,4,2 obtained by using the anticode A4,2 is
m a [2m−7,m,2m−1−4] LRC with locality r =2 which
(Gm)T 1
attains the bound (2).
We will show that by a simple modification of H with • The code Cm,5,2 obtained by using the anticode A5,2 is
elementary operations on its rows we obtain a parity-check a[2m−11,m,2m−1−6]LRCwithlocalityr =2which
matrix H(cid:48) such that every coordinate of the code will be attains the bound (2).
covered by a row of H(cid:48) of weight 3. We define a partition
of columns of H into the parts {H1,...,Hm} as follows. Proof:Toprovetheoptimalityoftheproposedcodeswe
The part H1 contains the first m columns, and the part Hi, apply the bound (2) with t = 1 and use the Plotkin bound
(see Thm. 1):
2 ≤ i ≤ m, corresponds to the columns which contain
I included in the rows which contain (Gi)T. (Note that For s = 3 we have 2+ko(2p)t(2m −7,2m−1 −2) ≤ 2+
if(mis)= m then H2 = ∅.) Let consider the xth coordinate (cid:106)log2(cid:106)2m−31−2(cid:107)(cid:107)≤2+(cid:4)log(2m−1−2)(cid:5)=2+m−2=m.
in Hi, 3 ≤ i ≤ m−1. There exists a row Ri in H with For s = 4 we have 2+k(2)(2m −10,2m−1 −4) ≤ 2+
x opt
nonzero entry in this coordinate. This row Rxi also contains (cid:106)log2(cid:106)2m−1−4(cid:107)(cid:107)=2+(cid:4)log(2m−1−4)(cid:5)=2+m−2=m.
i nonzero entries in the first m coordinates. Let Ri+1 be 2
a row in H such that its i+1 nonzero entries in thye first For s = 5 we have 2+ko(2p)t(2m −14,2m−1 −6) ≤ 2+
m coordinates contain the first i nonzero entries of Ri, and (cid:106)log2(cid:106)2m−1−6(cid:107)(cid:107)=2+(cid:4)log(2m−1−6)(cid:5)=2+m−2= m.
x 2
which also has one in the yth coordinate of Hi+1. Then the
coordinatesxinHiandyinHi+1arecoveredbyRi+Ri+1,
x y Remark 1. One can check that the codes C and C
the weight-3 row of H(cid:48). Note that for every x in Hi there is m,3,2 m,5,2
attain the Griesmer bound.
such y in Hi+1. To show that any coordinate in H1∪Hm
has locality 2, note that when we add the last row of H to Note that to apply our modification of a parity-check
eachoneofthemrowswhichcontaintherowsof(Gm−1)T, matrix in the proof for locality 2, the generator matrix of
a code obtained by the Farrell construction should contain code. Note that C is the augmented simplex code S with
m−1
columns of consecutive weights. Based on this observation, generator matrix:
we propose a generalization of the previous construction of
(cid:18) (cid:19)
an anticode and prove that the LRCs obtained from this 111...11
G= .
anticode have locality r =2 and attain the Griesmer bound. G
m−1
Theorem 5. Let S be a [2m−1,m,2m−1] simplex code,
m Toprovethatthelocalityis3,wenotethatallthecodewords
m≥4,andletG beitsgeneratormatrix.LetA ,
m t;2,3,...,t−1 inthedualcodeofC haveevenweight,andhencethelocality
3 ≤ t ≤ m, be an anticode such that its generator
r is an odd number. Clearly, r >1. We construct the parity-
matrix G consists of all the columns in Ft of weights in
A 2 check matrix H of C with all the rows of weight 4, and
{2,3,...,t−1}. We prepend m−t zeros to every column
of G to form the m×(cid:80)t−1(cid:0)t(cid:1) matrix whose columns are then every coordinate will be covered by a row of H of
A i=2 i weight 4, which by Lemma 1 implies that r =3. Recall that
deleted from G to obtain a generator matrix G for a new
m the dual code of S is a [2m−1,2m−1−m,3] Hamming
codeC .ThenC isa[2m−2t+t+1,m,2m−1−2t−1+2] m−1
m,t m,t code [10], and denote its generator matrix by Hm−1. We
LRC with locality r =2 which attains the Griesmer bound.
consider the construction of Hm−1 with all the rows of
Proof: First we prove that the anticode A has weight 3 from [1], where the rows have nonzero entries in
t;2,3,...,t−1
length 2t−t−2 and maximum weight 2t−1−2. Note that the positions (i,2j,i+2j), for 1≤j ≤m−2, 1≤i≤2j.
thegeneratormatrixofA canbeobtainedfromthe LetdenotebyHm−1 arowofHm−1 withnonzeroentriesin
t;2,3,...,t−1 i,2j
generator matrix G of the simplex code S by removing t the positions (i,2j,i+2j). The parity-check matrix H will
t t
columns of weight 1 and one column of weight t, and hence consist of 2m−1 −m−1 weight-4 rows Hm−1 +Hm−1,
1,2 1,2j
thelengthofAt;2,3,...,t−1is2t−t−2.Sinceallthecodewords 2 ≤ j ≤ m−2, and Him,2−j1 +Him+−1,12j, 2 ≤ j ≤ m−2,
inSt haveweight2t−1 andfromeachrowofGt weremoved 1≤i≤2j −2.
twoonestoobtainageneratormatrixforouranticode,where
To prove the optimality of the obtained code C we apply
all the rows in G have one of the removed ones in the same
t the bound (2) with t=1 and use the Plotkin bound:
place, it follows that the maximum weight of A
t;2,3,...,t−1
idsoeδs=no2tt−ch1a−ng2e.tHheenlecne,gtshinacnedptrheepemndaxinimguzmerowreoiwghsttoofGthAe 3+ko(2p)t(2m−1−1−4,2m−2−1)≤3+(cid:22)log2·(cid:22)2m−32−1(cid:23)(cid:23)
anticode, C is a [2m −2t +t+1,m,2m−1 −2t−1 +2]
m,t
code. The proof of locality is similar to the proof in Thm. 3.
To prove that C attains the Griesmer bound we have ≤3+(cid:4)log(2m−2−1)(cid:5)=3+m−3=m.
m,t
m(cid:88)−1(cid:24)2m−1−2t−1+2(cid:25) m(cid:88)−1 (cid:88)t−1
= 2i− 2i+(2+t−1)
2i
i=0 i=0 i=0 Remark 2. One can check that the code C from Thm. 5
attains the Griesmer bound.
=2m−1−2t+1+t+1=2m−2t+t+1,
which completes the proof.
B. LRCs based on Subspace Codes
In the following, we consider an anticode formed by
another modification of a simplex code and prove that the In this subsection we consider a construction of optimal
LRC obtained from this anticode attains the bound (2) and binary LRCs with the parity-check matrix formed by the
has locality 3. codewordsofasubspacecode.Inparticular,weareinterested
in a special kind of subspace codes, called lifted rank-metric
Theorem6. LetA betheanticodewithgeneratormatrix
m−1 codes [17], with the trivial distance 1 and the constant di-
G given by
A mension2.Moreprecisely,letconsidera22s−4×(2s−2s−2)
1 000...00 binary matrix Hs whose columns are indexed by the vectors
GA = 0... Gm−1 , itvnhe∈F2Fs2-sd\−im{20}e0.nvEsivo:envrayl∈rsouFwb2ss−opfa2c}Hesasnoidsf wtFhs2heocisnoecnitrdaoeiwnnecsdeavirneecFitnos2dr\eox{fe0ds0uvbchy:
0 2
a 2-dimensional subspace and then has weight 3 (note that
where G is the generator matrix for the simplex code the all-zero vector is not considered). It was proved in [17,
m−1
S . Let C be a code obtained by the Farrell construction Thm. 11] that the code Cs with the parity-check matrix Hs
m−1
based on the simplex code S and on the anticode A . isa[2s−2s−2,s,2s−2s−2]2s−2-quasi-cycliccode.Notethat
m m−1 2
ThenC isa[2m−1−1,m,2m−2−1]LRCwithlocalityr =3 Hs contains dependent rows.
which attains the bound (2).
Example 4. For s = 4, the matrix H4 has the following
Proof: Since all the codewords in S have the form. The four rows above this matrix represent the vectors
m−1
constant weight 2m−2, the maximum weight of A is whichindexthecolumnsofH4.Forexample,thefirstrowof
m−1
δ =2m−2+1.Then,bytheFarrellconstruction,C isa[2m− H4 corresponds to the subspace which contains the vectors
1−2m−1,m,2m−1−2m−2−1]=[2m−1−1,m,2m−2−1] {(0,1,0,0),(1,0,0,0),(1,1,0,0)}:
0000 1111 1111 TABLEI
1111 0000 1111 OPTIMALBINARYLRCWITHLOCALITYTWOANDTHREE.
0011 0011 0011
0101 0101 0101
[n,k,d] Localityr Reference
1000 1000 1000 [28,5,14] 2 Thm.4 m=5,s=3
0100 0001 0010 [25,5,12] 2 m=5,s=4
0010 0100 0001 [21,5,10] 2 m=5,s=5
0001 0010 0100 [60,6,30] 2 Thm.4 m=6,s=3
1000 0100 0100 [57,6,28] 2 m=6,s=4
0100 0010 0001 [53,6,26] 2 m=6,s=5
0010 1000 0010
[21,5,10] 2 Thm.5 m=5,t=4
0001 0001 1000
[38,6,18] 2 m=6,t=5
1000 0010 0010
[31,6,15] 3 Thm.6 m=6
0100 0100 1000 [63,7,31] 3 m=7
0010 0001 0100 [24,5,12] 2 Thm.7 s=5
01000010 10000001 00000011 [48,6,24] 2 s=6
0100 1000 0100
0010 0010 1000
0001 0100 0010 [3] A.Dimakis,K.Ramchandran,Y.Wu,andC.Suh,“Asurveyonnetwork
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Theorem 7. Let Cs be the linear code with the parity-check
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6925–6934,Nov.2012.
distanceare provedin [17,Thm. 11].Sinceevery rowin Hs
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opt 3
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