Bounds and Constructions for Linear Locally Repairable Codes over Binary Fields

0.4 MB·

by  Anyu Wang

Bounds and Constructions for Linear Locally
Repairable Codes over Binary Fields
Anyu Wang*, Zhifang Zhang†, and Dongdai Lin*
*StateKeyLaboratoryofInformationSecurity,InstituteofInformationEngineering,CAS,Beijing,China
†KLMM,NCMIS,AcademyofMathematicsandSystemsScience,UniversityofChineseAcademyofSciences
Emails:[email protected],[email protected],[email protected]
Abstract—For binary [n,k,d] linear locally repairable codes constructed via anticodes in [16] also meets (2) with equality.
(LRCs), two new upper bounds on k are derived. The first one Although these codes are optimal with respect to the C-M
applies to LRCs with disjoint local repair groups, for general
bound, their code length n increases exponentially as the
values of n,d and locality r, containing some previously known
dimensionkgrows,implyingpoorperformanceininformation
bounds as special cases. The second one is based on solving
an optimization problem and applies to LRCs with arbitrary rate. Alternatively, cyclic codes provides more desirable can-
structure of local repair groups. Particularly, an explicit bound didates for LRCs over small fields. In [5], binary LRCs with
7 is derived from the second bound when d ≥ 5. A specific r = 2 and d = 2,6,10 are constructed from primitive cyclic
1 comparisonshowsthisexplicitboundoutperformstheCadambe- codes.ThesecodesdonotattaintheC-Mbound,butareshown
0 Mazumdarboundfor5≤d≤8andlargevaluesofn.Moreover,
to be optimal under a structural assumption that the codeword
2 a construction of binary linear LRCs with d ≥ 6 attaining our
second bound is provided. coordinates are divided into disjoint local repair groups. The
n same method is adopted in [24] to generate binary LRCs with
a I. INTRODUCTION r =2,d=10 from nonprimitive cyclic codes. In [21], BCH-
J
Recently, locally repairable codes (LRCs) have attracted a type binary LRCs are constructed as the subfield subcodes
4
2 lot of attention due to their applications in distributed storage of optimal Reed-Solomon-Type LRCs. Besides, some other
systems.An[n,k,d]linearcodeiscalledanLRCwithlocality approaches for constructing LRCs that attain the Singleton-

] rifthevalueateachcoordinatecanberecoveredbyaccessing like bound (1) over small fields are also developed in [7],
T
at most r other coordinates. An LRC with small locality r [14].
I
. is preferred in practice as it greatly reduces the disk I/O Recently,twoupperboundstakingthefieldsizeintoaccount
s
c complexity in repairing node failures. Meantime, large values are derived in [8] for (r,δ)-LRCs. Since (r,δ)-LRCs [12]
[ of k and d are also desirable to ensure high level of storage contain LRCs as the special case of δ =2, these two bounds

2 efficiencyandglobalfaulttoleranceabilityrespectively.Much applytoLRCsaswell.Forδ =2,thefirstboundisequivalent
v workhasbeendonetowardexploringtherelationshipbetween totheSingleton-likebound(1),whilethesecondoneisalinear
9 the parameters n,k,d,r. The first trade-off is derived in [4], programming bound for LRCs with disjoint repair groups. On
8
i.e., theotherhand,asymptoticboundsontheparametersofLRCs
9
(cid:6)k(cid:7) are studied in [1], [20].
5 d≤n−k− +2, (1)
0 r Overall, most of the bounds derived so far for LRCs over
. which is also known as the Singleton-like bound for LRCs. particular finite fields either depend on undetermined parame-
1
0 Then various methods are developed to construct LRCs at- ters in coding theory, e.g., the C-M bound, or rely on solving
7 taining the bound (1), e.g., [11], [15], [19], [22]. Tightness of optimization problems under concrete code parameters, e.g.,
1 the singleton-like bound is discussed in [6], [17], and some the LP bound in [8]. And the constructions of binary LRCs
:
v improved bounds are derived in [13], [23]. with good parameters mostly restrict to specific values of d
i It can be seen the bound (1) does not care about the and r. Much work remains undone for LRCs over particular
X
field size. However, in practice LRCs over small finite fields, finitefields.Inthiswork,wefocusonlinearLRCsoverbinary
r
a especially those over binary fields, are preferred due to their fields.
convenience in implementation. The first trade-off taking
into consideration the field size is derived by Cadambe and A. Main Idea and Contribution
Mazumdar [2], i.e.,
For any [n,k,d] binary linear LRC C, our main idea for
k ≤ Min(cid:2)tr+ko(qp)t(n−(r+1)t,d)(cid:3), (2) deriving upper bounds on k is to consider a related sphere
t∈Z+
packing problem in a particular space, namely, the L-space.
where k(q)(n,d) is the largest possible dimension of an Specifically, an L-space of C is defined to be the dual of the
opt
[n,k,d]linearcodeoverF .Thistrade-offisusuallycalledthe linear space spanned by a minimum set of local parity checks
q
C-M bound, and is proved achievable by the binary Simplex with overall supports covering all coordinates. Actually, the
codes [2]. Another class of binary LRCs with r = 2,3 L-space can be viewed as an LRC which contains C as a
subcode. Then by applying the sphere packing bound in the where BV((cid:4)d−21(cid:5)) = (cid:12)(cid:12){v ∈V :wt(v)≤(cid:4)d−21(cid:5)}(cid:12)(cid:12), and V is
L-space, two upper bounds for C are derived. an L-space of C.
Firstly,assumingthecodeC hasdisjointlocalrepairgroups,
Proof: For any codeword c ∈ C, consider the ball of
we get an explicit bound (i.e. Corollary 3) on k for general radius (cid:4)d−1(cid:5) around c in V. Since C has minimum distance
values of n,d,r. Note that for r = 2 and special forms of 2
d, then these balls are non-overlapping. It follows that
n,d, upper bounds were also derived in [5], [24]. It turns out
our bound contains their results as special cases. (cid:88) (cid:4)d−1(cid:5)
B (c, )≤|V|, (4)
Secondly, for linear binary LRCs with arbitrary local repair V 2
groups,wederiveanupperbound(i.e.Theorem5)onk based c∈C
odniffiscoulvltintogsaonlvoeptthimisizoapttiiomnizpartoibolnempr.oAbllethmo,usgihmpitliifiscgateinoenraclalyn wthhaetrCe⊆BVV(,cs,o(cid:4)dw−2e1h(cid:5)a)v=e B(cid:12)(cid:12){v(c∈,(cid:4)Vd−:d1i(cid:5)s)t(=v,Bc)≤((cid:4)(cid:4)d−d−12(cid:5)1)(cid:5),}∀(cid:12)(cid:12)c. N∈oCte.
V 2 V 2
be done for d ≥ 5, and thus an explicit upper bound (i.e. Therefore (4) can be written as
Theorem 6) is derived. Through a specific comparison, we
showthisboundcanoutperformtheC-Mboundfor5≤d≤8 |C|·B ((cid:4)d−1(cid:5))≤|V|. (5)
andlargevaluesofn.Moreover,aclassofbinarylinearLRCs V 2
with d≥6 attaining this explicit bound is constructed. Since log |C|=k and log |V|=dim(V), then the Theorem
2 2
follows directly from (5).
B. Organization
The right hand side of (3) depends on the locality space
Section II defines the L-space, and derives our first bound V, so explicit bound can be derived from (3) if V is known.
(i.e. Corollary 3). Section III presents our second bound
In the following, we apply Proposition 2 to a special class of
(Theorem 6). Section IV gives the construction attaining our binary linear LRCs which has a clear L-space.
second bound. Section V concludes the paper.
A. Bound for LRCs with Disjoint Local Repair Groups
II. THEL-SPACEFORLRCS
For any vector v = (v1,...,vn) ∈ Fn2, let Supp(v) = satAissfysuinmge∪ClhaSsulpopca(hl pa)rit=y ch[nec],kswht(ih1,h)i2,=...r,h+il 1∈aCn⊥d
{[ni] ∈= [n{]1,:2,.v.i.,(cid:54)=n}.0F}oranadnywtt(wvo) v=ecto|Srsupup(,vv)|,∈whFen2re, Sru+pp1(h|ijn)∩ajnS=du1plp(=hij(cid:48))nij=. ∅Sufcohr 1an≤LjR(cid:54)=Cijjis(cid:48) ≤usul.alOlybvsiaoiudsltyo,
dist(u,v) denotes the hamming distance of u and v. Denote r+1
have disjoint local repair groups, which is widely adopted
by Span (u ,...,u ) the linear space spanned by a set of
vectors {2u ,1...,u }l over F . in constructions of LRCs, e.g., [11], [15], [19], [22]. Under
1 l 2
this assumption, the structure of the L-space V becomes quite
Let C be an [n,k,d] binary linear LRC with locality r.
simple. Then based on Proposition 2, we derive the following
Then for each coordinate i∈[n], there is a local parity check
upper bound.
h ∈C⊥ such that i∈Supp(h ) and wt(h )≤r+1. Note by
i i i
local parity checks we mean the codewords in the dual code Corollary 3. For any [n,k,d] binary LRC C with locality r
C⊥ with weight at most r+1. that has disjoint local repair groups, it holds
Definition 1. Let H ⊆ {h1,(cid:83)...,hn} be a set of local rn (cid:0) (cid:88) (cid:89)l (cid:18)r+1(cid:19)(cid:1)
parity checks of C such that Supp(h) = [n], and k ≤ −log . (6)
(cid:83)h∈H(cid:48)Supp(h) (cid:54)= [n] for any Hh(cid:48)∈(cid:40)H H. We call H an L- r+1 2 0≤i1+···+il≤(cid:98)d−41(cid:99)j=1 2ij
cover of C. Denote H=Span (H), then the dual space of H,
2
i.e., V ={v ∈Fn2 |v·h=0, ∀h∈H}, is called an L-space Proof: Note that H = {hi1,hi2,...,hil} is an L-
of C. cover of C, then V = Span (H)⊥ is an L-space of C. By
Proposition2,itsufficestodet2erminedim(V)andB ((cid:4)d−1(cid:5)).
Obviously, an L-cover H contains the minimum number Clearly it has dim(V) = rn . Note that the linVear s2pace
of local parity checks guaranteeing the locality r for all r+1
Span (H) has weight enumerator polynomial W (x,y) =
coordinates. H needs not be unique, neither does the L-space 2 H
(xr+1+yr+1)l. Then by the MacWilliams equality, (see e.g.,
V. Our proofs in this paper only depend on their existence
[9]), the weight enumerator polynomial of V is
which is ensured by the definition of LRCs. Since H only
contains partial parity checks of C, it follows that V also 1
W (x,y)= W (x+y,x−y)
defines an LRC containing C as a subcode. Investigating the V 2l H
structure of V may help us to study the code C. In the = 1((x+y)r+1+(x−y)r+1)l
following, we use the sphere-packing bound in the L-space 2l
(cid:18) (cid:19)
V, and obtain a connection between k,d and V. =(cid:0)(cid:88) r+1 xr+1−2iy2i(cid:1)l
2i
Proposition 2. For an [n,k,d] binary LRC C with locality r, i≥0
(cid:88)
it holds = A xn−2uy2u,
(cid:0) (cid:4)d−1(cid:5) (cid:1) u
k ≤dim(V)−log2 BV( 2 ) , (3) 0≤u≤n2
where A = (cid:80) (cid:81)l (cid:0)r+1(cid:1). Thus we have matrix defines a binary code C(cid:48). It can be proved C(cid:48) has the
u i1+···+il=u j=1 2ij following properties.
B ((cid:4)d−1(cid:5))=A +···+A Lemma 4. The shortened code C(cid:48) is an [N,K,D] binary
V 2 0 (cid:98)d−41(cid:99) linear LRC satisfying
l (cid:18) (cid:19)
(cid:88) (cid:89) r+1 (cid:1) (i) n≥N ≥2n−l(r+1), K ≥N −(n−k), D ≥d;
= ,
0≤i1+···+il≤(cid:98)d−41(cid:99)j=1 2ij (ii) Cw(cid:48)t(hha(cid:48)s)a≤n rL+-co1vearndHS(cid:48)u=pp({hh(cid:48)(cid:48)i1),∩..S.u,php(cid:48)il(}h(cid:48)su)ch=t∅haftor1a≤ll
and (6) follows directly. 1≤jij(cid:54)=j(cid:48) ≤n. ij ij(cid:48)
The sphere packing approach was also used in [5], [24] to
Proof: Since the shortening operation neither increases
deriveupperboundsonk forbinarylinearLRCswithdisjoint
theredundancynordecreasestheminimumdistance,(seee.g.,
local repair groups. However, their approach only applies to
[9]), then it has K ≥ N −(n−k) and D ≥ d. To show the
thecaseofr =2becauseitreliesonamapfrombinarylinear
LRCs with r =2 to additive F -codes. Our bound works for other statements, we suppose without loss of generality that
4
general values of n,d,r, especially containing the bounds in L=(cid:0)L(cid:48), L(cid:48)(cid:48)(cid:1),
[5], [24] as special cases. For example, suppose n = 2m −
1,d=6 and r =2, then Corollary 3 implies that where L(cid:48) ∈Fl2×N consists of the N columns that have weight
2n (cid:0)(cid:18)r+1(cid:19) (cid:18)r+1(cid:19)(cid:1) 1, and L(cid:48)(cid:48) ∈Fl2×(n−N) consists of the other (n−N) columns
k ≤ −log +l that have weight ≥ 2. By counting the number of 1’s in L,
3 2 0 2
we have
2n
= −log (1+n)
3 2 l(r+1)≥ the number of 1’s in L
2
= (2m−1)−m, ≥N +2(n−N).
3
Thus 2n−l(r+1) ≤ N ≤ n. Lastly, denote h(cid:48),...,h(cid:48) to
which coincides with the Theorem 1 in [5]. 1 l
be the rows of L(cid:48), then clearly {h(cid:48),...,h(cid:48)} is a set of parity
Another bound for LRCs with disjoint repair groups is the 1 l
checks of C(cid:48) such that 1 ≤ wt(h(cid:48)) ≤ r+1. Note that each
LP bound derived in [8]. Table 1 lists a comparison of the i
column of L(cid:48) has exactly one 1, therefore ∪l Supp(h(cid:48)) =
bound (7), the LP bound in [8] and the C-M bound (2) for i=1 i
[N] and Supp(h(cid:48))∩Supp(h(cid:48)) = ∅ for all 1 ≤ i (cid:54)= j ≤ n,
3 ≤ r ≤ 10, n = 3,d = 5. From the table we can see the i j
r+1 which completes the proof.
bound (6) is slightly weaker than the LP bound but tighter
Let V(cid:48) = Span (H(cid:48))⊥ be an L-space of C(cid:48), and denote
than the C-M bound (2). Nevertheless, the bound (6) has an 2
wt(h(cid:48) )=r +1forj ∈[l].Thenitfollowsdim(V(cid:48))=N−l,
explicit form and can be more easily implemented than the ij j
and by a deduction similar to that in Corollary 3 it has
other two bounds.
l (cid:18) (cid:19)
r 3 4 5 6 7 8 9 10 B ((cid:98)D−1(cid:99))= (cid:88) (cid:89) rj +1 .
V(cid:48) 2 2i
TheOuCr-Mboubnodun(d6)(2) 45 77 190 1123 1145 1178 1291 2223 0≤i1+···+il≤(cid:98)D4−1(cid:99)j=1 j
TheLPbound[8] 4 6 9 11 14 17 19 22 Applying Proposition 2 to the shortened LRC C(cid:48), we get
l (cid:18) (cid:19)
Table1 K ≤(N −l)−log (cid:0) (cid:88) (cid:89) rj +1 (cid:1). (7)
2 2i
j
0≤i1+···+il≤(cid:98)D4−1(cid:99)j=1
ThencombiningwithLemma4,wegetthefollowingtheorem.
III. NEWUPPERBOUNDFORBINARYLINEARLRCS
In this section we will remove the assumption of disjoint Theorem 5. For any [n,k,d] binary linear LRC with locality
local repair groups, and derive parameter bounds for linear r, it holds
binary LRCs with arbitrary local repair groups. Suppose C (cid:104) (cid:105)
k ≤n− Min l+log (Φ (r ,...,r )) , (8)
is an [n,k,d] binary linear LRC with locality r. Let H = l,r1,...,rl 2 l 1 l
{h ,...,h } ⊆ C⊥ be an L-cover of C. By shortening at
i1 il where
the coordinates that appear more than once in the supports of
l (cid:18) (cid:19)
hhais1,d.i.s.jo,hinitl,lowcealcraenpdaierrigvreofurposm. TChausshthoertepnroedblceomdeisCr(cid:48)ewduhciecdh Φl(r1,...,rl)= (cid:88) (cid:89) rj2+i 1
j
to that we discussed in last section. 0≤i1+···+il≤(cid:98)d−41(cid:99)j=1
Specifically, define L ∈ Fl×n to be a matrix whose rows and the ‘Min’ is taken over all integers l,r ,...,r such that
2 1 l
are the l local parity checks in H. Denote by N the number 
n ≤l≤ 2n ;
of columns in L that have weight 1. Let L(cid:48) be the matrix r+1 r+2
obtained from L by deleting the n−N columns of L that 0≤r1,...,rl ≤r; (9)
haveweightgreaterthan1.ThentakingL(cid:48) astheparitycheck r +···+r =2n−l(r+2).
1 l
Proof: From Lemma 4 it has K ≥N −(n−k),D ≥d. Proof: When d≥5, it has (cid:4)d−1(cid:5)≥1 and therefore
4
Then
l (cid:18) (cid:19)
Φ (r ,...,r )≥ (cid:88) (cid:89) rj +1
k ≤K−N +n l 1 l 2i
j
(a) n−l−log (cid:0) (cid:88) (cid:89)l (cid:18)rj +1(cid:19)(cid:1) =01≤+i1+(cid:18)·r··1++il≤11(cid:19)j=+1···+(cid:18)rl+1(cid:19).
≤ 2 2ij 2 2
(b) n−l−log (0Φ≤i(1r+·,·.·+.i.l,≤r(cid:98))D)4−,1(cid:99)j=1 Note that (cid:0)x+1(cid:1)= 1x(x+1) is a convex real-valued function
≤ 2 l 1 l 2 2
and it is required in (9) that r +···+r =2n−l(r+2), so
1 l
where (a) is from (7) and (b) holds because D ≥d. Note that (cid:18)1(cid:80)l (r +1)(cid:19)
the integers l,r1,...,rl satisfies Φl(r1,...,rl)≥1+l l j=12 j

n ≤l; 1
r+1 =1+ (2n−l(r+1))(2n−l(r+2))).
0≤r ,...,r ≤r; 2l
1 l
(cid:80)l r ≥2n−l(r+2). It follows from Theorem 5 that
j=1 j (cid:104) (cid:105)
k ≤n− Min l+log (Φ (r ,...,r ))
There are two cases. 2 l 1 l
l,r1,...,rl
n−Cars2+en21.:Oln>ther2+no2t.heOrnhathned,owneithhatnhde,rwesetrihcativoenk(9≤),nit−hals< ≤n−Mlin(cid:104)l+log2(cid:0)1+(2n−l(r+1)2)(l2n−l(r+2))(cid:1)(cid:105),
(cid:104) (cid:105) where the integer l satisfies n ≤l ≤ 2n according to (9).
Min l+log (Φ (r ,...,r )) r+1 r+2
2 l 1 l Let
l,r1,...,rl
2n (cid:16) (cid:17) 1
≤ +log Φ (0,...,0) f(l)=l+log (1+ (2n−l(r+1))(2n−l(r+2)))
r+2 2 r2+n2 2 2l
= 2n . be a function defined for integers l∈[ n , 2n ]. We claim
r+1 r+2
r+2
n rn rn
f(l)≥ +min{log (1+ ), }
So inequality (8) holds. r+1 2 2 (r+1)(r+2)
Case 2: l≤ 2n . In this case it has
r+2 for 2≤r ≤ n −2, then the theorem follows directly.
2
 Firstly, we show that f(cid:48)(cid:48)(l) ≤ 0 for 2 ≤ r ≤ n −2. Note
n ≤l≤ 2n ; 2
r+1 r+2 that
0≤r ,...,r ≤r; . (10)
1 l 80n4−(l2(r2+3r+2)−8n2)2−16l(n3(3+2r)−n2)
(cid:80)l r ≥2n−l(r+2). f(cid:48)(cid:48)(l)= ,
j=1 j l2(4n2+l2(2+3r+r2)+l(2−2n(3+2r)))2ln2
So we have then it suffices to prove g(l) ≤ 0 for 2 ≤ r ≤ n −2, where
2
(cid:104) (cid:105) g(l)=80n4−(l2(r2+3r+2)−8n2)2−16l(n3(3+2r)−n2).Since
k ≤n− Min l+log2(Φl(r1,...,rl)) , g(cid:48)(cid:48)(cid:48)(l) = −24l(r2 +3r+2)2 < 0, it has g(cid:48)(l) is a concave
l,r1,...,rl function with g(cid:48)( n ) = 4n2(4r+4−nr2) < 0 and g(cid:48)( 2n ) =
wsohaerneetchees‘sMaryinc’oisndtaitkieonnofoverro(p1t0im).iNziontge(t8h)atis2nth−atl((cid:80)r+l 2)r≥=0, 16n2(r2++2r+nr) >0r.+It1follows thra+t1 r+2
j=1 j
2n−l(r+2). Then the optimization can be restricted to the n 2n
g(l)≤max{g( ),g( )}
condition (9), and thus the theorem holds. r+1 r+2
For any given n,d,r, Theorem 5 gives an upper bound on n3(16(r+1)−n(r+2)2) 16n3(2(r+2)−n)
=max{ , }
the dimension k based on solving an optimization problem. (r+1)2 (r+2)2
However, solving the optimization problem is very difficult
≤0
in general since the objective function in (8) is nonlinear.
Nevertheless, it is still possible to simplify the bound (8) in for all 2≤r ≤ n −2.
2
some special cases. Next, we will derive an explicit upper According to f(cid:48)(cid:48)(l) ≤ 0, f(l) is a concave function, then
bound from Theorem 5 for d≥5 . we have
n 2n
f(l)≥min{f( ),f( )}
A. Explicit Bound for d≥5 r+1 r+2
n rn 2n
Theorem 6. For any [n,k,d] binary linear LRC with locality =min{ +log (1+ ), }
r such that d≥5 and 2≤r ≤ n −2, it holds r+1 2 2 r+2
2 n rn rn
= +min{log (1+ ), },
rn rn rn r+1 2 2 (r+1)(r+2)
k ≤ −min{log (1+ ), }. (11)
r+1 2 2 (r+1)(r+2) and therefore the theorem follows.
k≤
Since we focus on linear codes, then k is actually upper
40 □□□
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□□
□
of n. Specifically, Fig. 1 and Fig. 2 display comparisons of
n
the two bounds for r =3,d=5,10≤n≤60 and r =2,d= 20 30 40 50 60
8,60≤n≤110 respectively. Moreover in Table 2, based on
Fig. 1. A comparison of the C-M bound and the bound in Theorem 6 for
a detailed calculation of the two bounds for 2 ≤ r ≤ 5 and r=3,d=5,10≤n≤60.
2≤n≤250, we list the tipping points of n’s that the bound
(11) is tighter than the C-M bound thereafter. k≤
□
□□●
r=2 r=3 r=4 r=5 65 □□ TheC-Mbound □□●□●□●●●
□□●
d=5 n≥19 n≥26 n≥34 n≥43 60 ●● Ournewbound(10) □□●□●●●
□●
d=6 n≥23 n≥31 n≥40 n≥50 □□●□●●
□□●●
d=7 n≥31 n≥41 n≥84 n≥125 55 □●
□□●●
d=8 n≥50 n≥70 n≥145 n≥228 □□●□●●
□●
50 □□●●
□●
Table2 □□●□●●
□□●●
45 □●●
□□●
□●●
□□●
□●●
For d≥9, it can be checked that the bound (11) is inferior 40 □●□●□●
□□●
to the C-M bound. A reason causing this disadvantage is that □●●
□●□●
in this case it has (cid:4)d−1(cid:5)≥2 while the bound (11) is derived 35□●□●□● n
by lower-bounding Φ4 by its value in the case (cid:4)d−1(cid:5)=1. If 70 80 90 100 110
l 4
we use a better lower bound of Φ (r ,...,r ) instead of that Fig. 2. A comparison of the C-M bound and the bound in Theorem 6 for
l 1 l
r=2,d=8,60≤n≤110.
used in the proof of Theorem 6, an upper bound tighter than
(11) could be expected. However, we can not get an explicit
bound in this case yet since the corresponding optimization
of B for 1 ≤ i ≤ 2s−1. Then the binary linear LRC is
problem is still very complicated. 22t−1
constructed below.
IV. CONSTRUCTIONATTAININGTHEUPPERBOUND
Construction 1. Define C to be a binary linear code with the
In this section, we give a new construction of binary linear
parity check matrix
LRCs. The code has minimum distance d≥6, and attains the
upper bound (11) in Theorem 6. (cid:18)L L ... L (cid:19)
H = 1 2 l ,
TheconstructionreliesontwomatricesAandB definedas H H ... H
1 2 l
follows. Suppose s and t are two positive integers such that
2t | s and s ≥ 2. Let A be a binary matrix of size 2t×2t where l = 2s−1, and for 1 ≤ i ≤ l, L is an l×(2t +1)
2t 22t−1 i
such that any 4 columns of A are linearly independent. For matrix whose i-th row is the all-one vector and the other
t ≤ 2, A can be chosen as the identity matrix. For t ≥ 3, A rows are all-zero vectors, H is an s × (2t + 1) matrix
i
is a parity check matrix of a [2t,2t−2t,5] binary code which over F whose columns are binary expansions of the vectors
2
can be constructed from nonprimitive cyclic codes of length {0,a β ,a β ,...,a β }.
1 i 2 i 2t i
2t +1 (see e.g., [3]). We give a detailed construction of A
in Appendix A. Define B to be a matrix whose columns are
Example 1. Suppose s=4 and t=1. Then we can choose
all nonzero s-tuples from F with first nonzero entry equal
2t 22t
to 1. Then B is actually a parity check matrix of a 22t-ary (cid:18)1 0(cid:19) (cid:18) 1 1 1 1 0(cid:19)
Hamming code, and the size of B is s × 2s−1. A= 0 1 ∈F22×2, B = ω2 ω 1 0 1 ∈F24×5,
2t 22t−1
By fixing a basis of F over F , each vector in F2t can
22t 2 2
be written as an element in F22t and vice versa. We denote whereω isaprimitiveelementinF4 suchthatω2+ω+1=0.
by a1,...,a2t ∈ F22t the 2t elements corresponding to the Fixing a basis {1,ω}, the two columns of A can be written
columnsofA,anddenotebyavectorβ ∈F2st theithcolumn as two elements in F , i.e., a = (1,ω) · (cid:0)1(cid:1) = 1,a =
i 22t 4 1 0 2
(1,ω)·(cid:0)0(cid:1)=ω. Note that β =( 1 ), then We claim (cid:6)log (1+ rn)(cid:7) = s and (cid:6) rn (cid:7) ≥ s for all
1 1 ω2 2 2 (r+1)(r+2)
(cid:18) (cid:19) (cid:18) (cid:19) s,t satisfying 2t | s, s ≥ 2 except s = 4,t = 1. Then the
1 ω 2t
α β = , α β = . claim implies that k ≤ rn −s, and therefore C is optimal.
1 1 ω2 2 1 1 r+1
To show (cid:6)log (1+ rn)(cid:7)=s, note that
2 2
Byexpanding{0,α β ,α β }⊆F2 intobinaryvectorswith
1 1 2 1 4 rn 2t 2s−1
respect to the basis {1,ω}, we get log (1+ )=log (1+ · ).
2 2 2 2t−1 2
 
0 1 0
H1 =00 01 11. Since 1< 2t2−t1 ≤2, we have
2t 2s−1
0 1 0 2s−1 <1+ · ≤2s.
2t−1 2
The other H ’s can be computed similarly, so we have
i It follows that (cid:6)log (1 + rn)(cid:7) = s. It remains to show
111  (cid:6) rn (cid:7)≥s. W2hen t=21, it has s>4 and
 111  (r+1)(r+2)
 
 111  rn 2t 2s−1
  = ·
 111  (r+1)(r+2) (2t+1)(2t+2) 2t−1
 
H = 111. 1
010 010 010 010 000 = 6(2s−1)
 
001 001 001 001 000 ≥s.
 
011 001 010 000 010
010 011 001 000 001 When t≥2, it has
It can be verified that any 5 columns of H are linearly rn 2t 2s−1
= ·
independent. So H defines an [n = 15,k = 6,d ≥ 6] binary (r+1)(r+2) (2t+1)(2t+2) 2t−1
LRC with locality r = 2. Substituting n = 15,d = 6,r = 2 2t 1 s(24t−1)
into the C-M bound (2) yields k ≤ 6, so this binary linear (≥a) (2t+1)(2t+2) · 2t−1 · 4t
LRC is optimal with respect to the C-M bound.
2t 22t+1
= · ·s
Theorem 7. The code C obtained from Construction 1 is an 2t+2 4t
brin=ary2tl.inMeaorreLoRvCer,wCithatntai=ns22tsth−−e11,ukpp≥errrb+no1u−nds,(1d1≥) fo6raanldl (≥b) s,
positive integers s,t satisfying 2t | s and s ≥ 2 except the where (a) holds since 2s−1 ≥ 24t−1, which is a consequence
2t s 4t
case s=4,t=1. of s≥4t, and (b) holds since 2t ≥ 1 and 22t+1≥8t for
2t+2 2
t≥2.
Proof: Since the values of n,k,r can be determined
easily, we focus on proving d ≥ 6. Note that the sum of the
V. CONCLUSIONS
firstlrowsofH isanall-onevector,sotheminimumdistance
of C must be even. Therefore it suffices to show that d ≥ 5. We introduce the concepts of L-covers and L-spaces for
Suppose to the contrary that there exists a codeword c ∈ C LRCs. By using the sphere-packing bound in the L-spaces,
such that Hcτ = 0, 1≤wt(c)≤4. Denote c=(c ,...,c ), we derive new upper bounds on the dimension k for binary
1 l
where c ∈ F2t+1 for i ∈ [l]. It can be deduced from the linear LRCs. Two explicit bounds are given respectively for
i 2
definition of L that wt(c ) is even, ∀i ∈ [l]. So there are LRCswithandwithouttheassumptionofdisjointlocalrepair
i i
at most two nonzero vectors in c ,...,c . Without loss of groups. Comparing with previously known bounds for LRCs
1 l
generality,wesupposec =···=c =0and1≤wt(c ,c )≤4. over particular finite fields, our bounds present an explicit
3 l 1 2
Then by Hcτ = 0 it has H cτ + H cτ = 0. Denote form, generalize previous results to general cases, and out-
1 1 2 2
c =(x ,x ,...,x ) and c =(y ,y ,...,y ), we have perform previous bounds in some cases. Moreover, a class of
1 0 1 2t 2 0 1 2t
binary codes attaining our second bound are also designed. A
(x1a1+···+x2ta2t)β1+(y1a1+···+y2ta2t)β2 =0. further investigation into the L-spaces are expected to bring
Since β and β are linearly independent over F , it must more results in studying binary LRCs.
1 2 22t
has (x a +···+x a )=(y a +···+y a )=0, which
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