Multi-Erasure Locally Recoverable Codes Over Small Fields For Flash Memory Array

0.69 MB·

by  Pengfei Huang

Multi-Erasure Locally Recoverable Codes Over Small
Fields For Flash Memory Array
Pengfei Huang∗, Eitan Yaakobi†, and Paul H. Siegel∗
Electrical and Computer Engineering Dept., University of California, San Diego, La Jolla, CA 92093 U.S.A
∗
Computer Science Dept., Technion – Israel Institute of Technology, Haifa 32000, Israel
†
pehuang,psiegel @ucsd.edu, [email protected]
{ }
Abstract—Erasure codes play an important role in storage sys- For the code construction, we focus on constructing ME-
temstopreventdataloss.Inthiswork,westudyaclassoferasure LRCs over small fields. We propose a general construction
codescalledMulti-ErasureLocallyRecoverableCodes(ME-LRCs) method from generalized tensor product codes [13], [22]. In
for flash memory array. Compared to previous related works, we
7 contrast to [5], our construction is more flexible in the sense
focus on the construction of ME-LRCs over small fields. We first
1 developupperandlowerboundsontheminimumdistanceofME- that we do not require field size q > max ρ,n0 , and it can
0 LRCs.Theseboundsexplicitlytakethefieldsizeintoaccount.Our evengeneratebinaryME-LRCs.For2d0 >d{,wew}illshowthat
2
main contribution is to propose a general construction of ME- our construction gives optimal ME-LRCs over Fq, with respect
n LRCs based on generalized tensor product codes, and study their to our upper bound on the minimum distance.
a erasure-correctingproperty.Adecodingalgorithmtailoredforera-
J sure recovery is given. We then prove that our construction yields There are other related works. If the local code 0 of an
C
optimal ME-LRCs with a wide range of code parameters. Finally, ME-LRC is a single parity-check code, then the ME-LRC is a
2
2 we present several families of ME-LRCs over different fields. Locally Repairable Code (LRC) [7], [14], [15], [19], which is a
class of codes where a single erased symbol can be recovered

] I. INTRODUCTION byaccessingrothersymbols,andhereriscalledthelocalityof
T
Recently, erasure codes with both local and global erasure- this code. LRCs over small fields were studied in [6], [8], [10]–
I
. correcting properties have received considerable interests [5], [12], [20], [24]. In [16], (r,δ) LRCs were proposed. A code
s C
c [7], [14]–[16], [19], thanks to their promising application in s- is called an (r,δ) LRC if for every coordinate, there exists a
[ torage systems. The idea behind is when a few erasures occur, punctured code (i.e., a repair set) of with support containing
1 these erasures can be corrected fast using only local parities. this coordinate, whose length is at mCost r+δ 1, and whose
−
v If the number of erasures exceeds the local erasure-correcting minimum distance is at leastδ. ME-LRCs studied in this paper
0 capability, then the global parities are invoked. canbeseenas(r,δ)LRCswithdisjointrepairsets.However,the
1
In this paper, we consider this kind of erasure codes with previous existing constructions of (r,δ) LRCs require relatively
1
both local and global erasure-correcting capabilities for a ρ large field size q, which is at least as large as the code length
6
×
0 n0 flash memory array [3], [5], where each row contains some [16], [18], [19]. A very recent work [1] gives explicit construc-
1. localparities,andadditionalglobalparitiesaredistributedinthe tions of (r,δ) LRCs with disjoint repair sets over field Fq from
0 array. More specifically, let us give the formal definition of this algebraic curves, whose repair sets have size r+δ 1 = √q
−
7 class of erasure codes as follows. or r+δ 1 = √q+1. In contrast, our construction is more
−
1 flexible, and can generate codes with field size q even smaller
v:oDfeρfinitniona1rr.aCysonsusicdherthaatc:odeC overafinitefieldFq consisting thantherepairsetsize.Inaddition,aPartialMDS(PMDS)code
i × 0 [4] can also be seen as an ME-LRC with certain constraints.
X 1) Eachrowineacharrayin belongstoalinearlocalcode
C C0 ThegoalofthispaperistostudyME-LRCsoversmallfields.
ar withlengthn0andminimumdistanced0overFq. The remainder of the paper is organized as follows. In Sec-
2) Readingthesymbolsof row-wise, isalinearcodewith
C C tion II, we study field size dependent upper and lower bounds
lengthρn0,dimensionk,andminimumdistancedoverFq. forME-LRCs.InSectionIII,weproposeageneralconstruction
Then,wesaythat isa(ρ,n0,k;d0,d)qMulti-ErasureLocally of ME-LRCs. The erasure-correcting property of these codes is
C
RecoverableCode(ME-LRC). studied and an erasure decoding algorithm is also presented. In
Thus, a (ρ,n ,k;d ,d) ME-LRC can locally correct d 1 SectionIV,wegiveseveralfamiliesofME-LRCsoverdifferent
0 0 q 0
erasuresineachrow,andisguaranteedtocorrectatotalofd−1 fields. Section V concludes the paper.
−
erasures anywhere in the array. Throughoutthepaper,weusethenotation[n]todefinetheset
Our work is motivated by a recent work by Blaum and Het- 1,...,n . For a length-n vector v over Fq and a set [n],
{ } I ⊆
zler [5]. In this work,the authors studied ME-LRCs where each the vector v denotes the restriction of the vector v to coordi-
row is a maximum distance separable (MDS) code, and gave nates in theIset , and wq(v) represents the Hamming weight
I
code constructions with field size q > max ρ,n0 from inte- of the vector over Fq. The transpose of a matrix H is written
{ }
grated interleaving codes [9]. As in Definition 1, we generalize as HT. A linear code over Fq of length n, dimension k, and
the definition of the codes studied in [5] such that each row is minimum distance d will be denoted by [n,k,d] , whose redun-
q
notnecessaryanMDScode.Moreover,wegiveupperandlower dancyisn k.Foracodewithonlyonecodeword,itsminimum
−
bounds on the minimum distance of ME-LRCs. distance is defined as .
∞
II. UPPERANDLOWERBOUNDSFORME-LRCS Proof: See Appendix A.
In this section, we study field size dependent upper and low-
er bounds on the minimum distance of ME-LRCs. The upper III. ME-LRCSFROMGENERALIZEDTENSORPRODUCT
bound derived here will be used to prove the optimality of our CODES:CONSTRUCTIONANDDECODING
construction for ME-LRCs in the following sections.
Tensor product codes, first proposed by Wolf in [22], are a
Now, we give an upper bound on the minimum distance of a
familyofbinaryerror-correctingcodesdefinedbyaparity-check
(ρ,n ,k;d ,d) ME-LRC. The shortening technique employed
0 0 q matrix that is the tensor product of the parity-check matrices of
here was previously used for bounds for low-density parity-
two constituent codes. Later, they were generalized in [13]. In
check (LDPC) codes [2] and locally repairable codes [6].
this section, we first introduce generalized tensor product codes
Let d(q)[n,k] denote the largest possible minimum distance
opt over Fq. Then, we give a general construction of ME-LRCs
of a linear code of length n and dimension k over Fq, and let from generalized tensor product codes. The minimum distance
k(q)[n,d] denote the largest possible dimension of a linear code oftheconstructedME-LRCsisdetermined,andacorresponding
opt
of length n and minimum distance d over Fq. decoding algorithm tailored for erasure correction is proposed.
Lemma2. Forany(ρ,n0,k;d0,d)q ME-LRC ,theminimum A. Generalized Tensor Product Codes over Fq
C
distancedsatisfies We start by presenting the tensor product operation of two
d606x6m⌈iknk∗⌉−1{d(oqp)t[ρn0−xn0,k−xk∗]}, (1) cmoadtericweisthHle′nagnthd nH′′′a.nLdedtimHe′nbsieonthne′p−arvityo-vcehreFckq.mTahterixmaotfrixa
andthedimensionsatisfies H′ can be considered as a v (row) by n′ (column) matrix over
k606x6m⌈iknk∗⌉−1{xk∗+ko(qp)t[ρn0−xn0,d]}, (2) FFqqv.orLeats Ha′1be(rotwhe) vbeyctnor′ (Hco′lu=mn[h)′1mha′2trix··o·fhe′nl′e]m, wenhtesrefrohm′j,
where k∗ =ko(qp)t[n0,d0]. m1a6trixj 6of an′c,oadreeoeflelmenegntthsℓofanFdqvd.imLeetnsHio′n′ bℓe thλeopvaerritFy-qcvh.eWcke
Proof: For the case of x = 0, it is trivial. For 1 6 x 6 denote H′′ by −
k 1, x Z+, let represent the set of the coordinates of
t⌈hke∗⌉fi−rst x ro∈ws in theIarray. Thus, = xn . First, consider h′′ h′′
|I| 0 11 ··· 1ℓ
tkhe, wcohdiechCsIati=sfi{escIk :6c ∈xk∗C.}Twhehno,sweedicmonensisdioernthisedceondoeted0b=y H′′ = ... ... ... ,
Ms{iIczo[eρrneo0o]fv\Iethr,:etchcIeod=mein0CimI0aInuidmscqdk∈i−stkCaIn}ca.enSddiˆnoictfeitsthheaeccloionddeeearC0coiissdelaitnaelseaCarw,sItethldle., where hi′′j, 16i6λand 1hλ′6′1 j6···ℓ, ahreλ′′ℓelements of Fqv.
C The tensor product of the matrices H′ and H′′ is defined as
i.e., dˆ>d. I
Thus, we get an upper bound on the minimum distance d, h′′ H′ h′′ H′
11 ··· 1ℓ
d6dˆ6do(qp)t[ρn0−|I|,k−kI] HTP = H′′ H′ = ... ... ... ,
6d(oqp)t[ρn0−xn0,k−xk∗]. ⊗  hλ′′1H′ ··· hλ′′ℓH′ 
Similarly, we also get an upper bound on the dimension k, where h′′H′ = [h′′h′ h′′h′ h′′h′ ], 1 6 i 6 λ and 1 6
ij ij 1 ij 2 ··· ij n′
k−kI 6ko(qp)t[ρn0−|I|,dˆ]6ko(qp)t[ρn0−xn0,d]. tjo6theℓ,rualneds othfempurlotidpulicctastioofnefloermeelenmtseanrtes ocvaelcruFlaqtve.dTahcecmoradtirnixg
Therefore, we conclude that HTP is considered as a vλ n′ℓ matrix of elements from Fq.
×
k6ko(qp)t[ρn0−xn0,d]+kI 6ko(qp)t[ρn0−xn0,d]+xk∗. sorOpurrocdouncsttrcuocdtieosn[o1f3]M. DE-eLfiRneCsthies bmaasetrdiceosn Hge′nearnadlizHed′′tefonr-
i i
An asymptotic lower bound for ME-LRCs with local MDS i = 1,2,...,µ as follows. The matrix Hi′ is a vi×n′ matrix
codes was given in [1]. Here, by simply adapting the Gilbert- over Fq such that the (v1+v2+···+vi)×n′ matrix
Varshamov (GV) bound [17], we have the following GV-like H′
1
lower bound on ME-LRCs of finite length without specifying
 H′ 
local codes. Bi = .2
 .. 
Lemma3.A(ρ,n0,k;d0,d)qME-LRC exists,if  
C  H′ 
<qdi∑ρ=−(02n0(−ρ⌈l(ongq0−∑dj⌈=0l−0o2g(qn0(j−∑1)d(j=q0−−012)(ijn0⌉−)j−1k).(q−1)j)⌉)−1)(q−1)i micsoadaterpi,xawroihtvyee-rrcehFdeqc′1vki<,mwadht′2ricix<hoi·sf·aa·n<p[anrd′i,ti′y.ni-T′c−hheecvkm1−amtravixt2ri−xHi′o·′f·i·sa−nav[λℓii,,dℓ×i′−]ℓq
( ) (3) λi,δi]qvi code.
We define aµ-level generalized tensor product code over Fq 1) The parity-check matrix H1′′ = Iℓ×ℓ, i.e., an ℓ×ℓ identity
as a linear code having a parity-check matrix in the form of the matrix.
followingµ-level tensor product structure 2) The matrices Hi′ (or Bi), 16i6µ, and H′j′, 26 j6µ, are
H′′ H′ chosen such that dµ′ 6δjd′j 1, for j =2,3,··· ,µ.
1 1 Step 2: Generateaparity-c−heckmatrix H accordingto(4)with
H = H2′′⊗.. H2′ . (4) the matrices Hi′ and Hi′′, for i = 1,2,...,µ. The constructed
 ⊗.  code corresponding to the parity-check matrix H is referred to
 Hµ′′ Hµ′  as CA. (cid:4)
 
dWimeLeedntesnuioostngeiivstehkiastn=ceoxdnaetm−bpyl∑eCµio=µGf1TavP2i.λ-lI⊗iet.svellegnegntheriasliznetd=tenns′oℓrapnroddtuhcet rTdahm=eeodtreµ′er.smρ5.=Thℓe,cno0de=CAn′i,ska(=ρ,nn′0ℓ,k−;d∑0µi,=d1)qviMλiE,-dL0R=Cwdi′1t,hapnad-
code CG2TP over F2. ρ, n ,Pkro,oafn:dAdccocradninbgeteoasCiloynsdtertuecrtmioinneAd.,Itnhethceodfoellpoawrainmge,tweres
0 0
Example1. Let H1′ =[1 1 1 1 1 1 1] over F2, and prove that d =dµ′ .
0 0 0 1 1 1 1 Since δ1 = (H1′′ is the identity matrix) and dµ′ 6δidi′ 1
H2′ = 0 1 1 0 0 1 1 for all i =2,3,...,µ, from Theorem 4, d>dµ′ . −
[ 1 0 1 0 1 0 1 ]
Now, we sho∞w that d 6 dµ′ . For i = 1,2,...,µ, let Hi′ =
over F2. Let H1′′ be a 3×3 identity matrix [h′1(i),...,h′n′(i)] over Fqvi, and let [h′1′1(i),...,hλ′′i1(i)]T over
1 0 0 Fqvi bethefirstcolumnof Hi′′.Sincethecodewithparity-check
H1′′ = 0 1 0 matrix Bµ has minimum distance dµ′ , there exist dµ′ column-
[ 0 0 1 ] s of Bµ, say in the set of positions J = {b1,b2,...,bdµ′ },
overF2,and H2′′ =[1 1 1] overF8.Hence,inthisconstruction, which are linearly dependent; that is, ∑j∈Jαjh′j(i) = 0, αj ∈
we have the following parameters: n′ = 7, ℓ = 3, v1 = 1, Fq, for i = 1,2,...,µ. Thus, we have ∑j Jαjh′p′1(i)h′j(i) =
v =3,λ =3,λ =1, d =2, d =4,δ = ,andδ =2. ∈
T2he binary1 parity-c2heck ma′1trix H o′2f the 2-1level tensor p2roduct h′p′1(i) ∑j Jαjh′j(i) = 0, for p = 1,2,...,λi and i =
∈
code CG2TP is ∞ 1,2,..(.,µ. That is, t)he columns in positions b1,b2,...,bdµ′ of
H are linearly dependent.
H′′ H′
H = 1 1 C. Erasure Decoding of ME-LRCs
[ H2′′⊗H2′ ]
We present a decoding algorithm for the ME-LRC from
1 1⊗1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CA
ConstructionA,tailoredforerasurecorrection.Thedecodingal-
0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0
  gorithmforerrorcorrectionforgeneralizedtensorproductcodes
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
= . can be found in [13].
 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 
  Let the symbol ? represent an erasure and “e” denote a de-
 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 1 1 
 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1  coding failure. The erasure decoder DA : (Fq ∪{?})n′ℓ →
  “e” for an ME-LRC consists of two kinds of com-
  CA∪{ } CA
The code length is nt = n′ℓ = 21 and the dimension is kt = ponentdecodersDi′ andDi′′ fori =1,2,...,µdescribedbelow.
nt−∑i2=1viλi =15. The minimum distance is dt =4. (cid:3) a) First, the decoder for a coset of the code Ci′ with parity-
check matrix B, i =1,2,...,µ, is denoted by
By adapting Theorem 2 in [13] from the field F2 to Fq, we i
pdoTrifhroeCedcoµGutrlcTyetPmchooa4ddvvt.eeer>TCtFhhµGmeqeT.Pmifnooi{lnvlδioem1wr,uFiδnm2qgdsd′1ath,itsiδesto3afidnree′2csm,e..do.tn,oδtfµhdeaµ′mg−e1inn,eidmrµ′au}lim.zeddistetannscoer swvcu′ehrc∈eitcosDh,CrDii′′uy+:s′i′(,e(Fsayeqn′to,dh∪snea′{)tf?lhoe=}elnl)goencwtn′h′,t×i-rn∑iigfe(ijsF=ydqw1′ec∪viatoghj{drsie?vyne}angs)ldu∑rwreuiojs=ilmte1hisvne:jevFxf→eoaqcrc,tt(oawlFyrhlqseoe∪′nrngewe{tiht?cth-h}oeno)d′unve′tiwenecoprtrouad-rt
Proof: See Appendix B. e is a coset leader determined by both the code Ci′ and the syn-
B. Construction of ME-LRCs drome vector s′; otherwise, Di′(y′,s′) = y′. Therefore, if the
Now, we present a general construction of ME-LRCs based length-n′ input vector y′ is a codeword in Ci′ +e with di′−1
or less erasures and the syndrome vector s is not erased, then
′
on generalized tensor product codes.
the decoder ′ can return the correct codeword.
Construction A Di
b) Second, the decoder for the code ′′ with parity-check
Step 1: Choosevi×n′ matricesHi′ overFq andλi×ℓmatrices matrix H′′, i =1,2,...,µ, is denoted byCi
Hi′′ over Fqvi, for i = 1,2,...,µ, which satisfy the following i
two properties: Di′′ : (Fqvi ∪{?})ℓ → (Fqvi ∪{?})ℓ
which uses the following decoding rules: for a length-ℓ input 2) Bµ is the parity-check matrix of an [n′,n′ −∑µi=1vi,dµ′ ]q
vector y′′, Di′′(y′′) = c′′, if y′′ agrees with exactly one code- code with d(oqp)t[n′,n′−∑µi=1vi]= dµ′ .
word c′′ ∈ Ci′′ on the entries with values in Fqvi; otherwise, 3) The minimum distances satisfy dµ′ 62d′1.
Di′′(y′′) = y′′. Therefore, if the length-ℓ input vector y′′ is a 4) Hi′′ is an all-one vector of length ℓ over Fqvi, i.e., the parity-
codeword in ′′ with δ 1 or less erasures, then the decoder check matrix of a parity code with minimum distance δ = 2,
Ci i− i
′′ can successfully return the correct codeword. for all i =2,...,µ. (cid:4)
Di
The erasure decoder for the code is summarized in Theorem7.Thecode fromConstructionAwithDesignIis
Algorithm 1 below. LetDthAe input word oCfAlength n′ℓ for the a(ρ= ℓ,n0 = n′,k =CnA′ℓ−v1ℓ−∑µi=2vi;d0 = d′1,d = dµ′ )q
decoder be y = (y ,y ,...,y ), where each component ME-LRC,whichisoptimalwithrespecttothebound(1).
DA 1 2 ℓ
yi ∈ (Fq ∪{?})n′, i = 1,...,ℓ. The vector y is an erased Proof: From Theorem 5, the code parametersρ, n0, k, d0,
version of a codeword c =(c1,c2,...,cℓ)∈CA. and d can be determined. We have k∗ =ko(qp)t[n′,d′1]=n′−v1.
Setting x=ℓ 1, we get
Algorithm 1: Decoding Procedure of Decoder DA −
IO1n.upLtupettu:tsr:1jecc=oeidv0ee,wdfoowrrdojrcd=∈y1C,=A2,(o.yr.1.a,,yℓd.2e,c.o.d.i,nygℓf)a.ilure indicator “e”. d66d0(o6qpx)t6m[ℓ⌈niknk∗′−⌉−1(ℓ{−d(oq1p)t)[nρ′n,0k−−x(ℓn0−,k1−)k∗x]k∗]}
µ
2. cˆ =(cˆ1,...,cˆℓ)= D1′(y1,s11),...,D1′(yℓ,s1ℓ) . =d(oqp)t[n′,n′− ∑vi]= dµ′ .
3. Let F ={j ∈[ℓ]: c(ˆj contains ?}. ) i=1
4. For i =2,...,µ This proves that achieves the bound (1).
CA
If = , do the following steps; otherwise go to step 5.
B. Explicit ME-LRCs from Construction A
• F ̸ ∅
• (si1,··· ,siℓ)= Di′′ cˆ1Hi′T,...,cˆℓHi′T . WegiveseveralexplicitME-LRCsoverdifferentfields.From
cˆ = ′ cˆ ,(s1,..(.,si) for j ;)cˆ remains the same Construction A, the local codes should have a nested structure
• j Di j j j ∈F j
and large minimum distance. Reed-Solomon codes and BCH
for j [(ℓ] . )
∈ \F codes are such codes. More generally, some cyclic codes and
Update = j [ℓ]: cˆ contains ? .
• F { ∈ j } algebraic geometry codes [21] also have these properties. In the
end
5. If = , let c =cˆ and output c; otherwise return “e”. following,someexplicitME-LRCsareconstructedoverF2,F3,
F ∅ and F4.
In Algorithm 1, we use the following rules for operations 1) ME-LRCs with local extended BCH codes over F2
which involve the symbol ?: 1) Addition +: for any element From the structure of BCH codes [17], there exists a chain
γ∈ Fq∪{?}, γ+? =?. 2) Multiplication ×: for any element of nested binary extended BCH codes: C3 = [2m,2m −1−
γn v∈ecFtoqr∪x{,?x}\∈{0(}F,qγ∪×{??}=)?n,, acnodnta0i×ns?a=n e0n.tr3y)?I,f tahelnenxgthis- 3mm,4,8]2].2 ⊂ C2 = [2m,2m−1−2m,6]2 ⊂ C1 = [2m,2m−1−
considered as the symbol ? in the set Fqn ∪{?}. Similarly, the Let the matrices B1, B2, and B3 be the parity-check matrices
symbol ? in the set Fqn ∪{?} is treated as a length-n vector of C1, C2, and C3, respectively.
whose entries are all ?.
Example2. For µ = 3, in Construction A, we use the above
Theorem6.ThedecoderDAfora(ρ,n0,k;d0,d)qME-LRCCA matrices B1, B2, and B3. We also choose H2′′ and H3′′ to be the
cancorrectanyd 1erasuresinthearray,andadditionalrows,
− all-one vector of length ℓ over F2m.
eachofwhichhaslessthanorequaltod0−1erasures. From Theorem 5, the corresponding (ρ,n0,k;d0,d)2 ME-
Proof: See Appendix C. LRC has parameters ρ = ℓ, n = 2m, k = 2mℓ (m+
CA 0 −
1)ℓ 2m, d = 4, and d = 8. This code satisfies the require-
0
IV. SEVERALFAMILIESOFME-LRCSOVERSMALLFIELDS men−ts of Design I. Thus, from Theorem 7, it is optimal with
In this section, we study the optimality of Construction A, respect to the bound (1). (cid:3)
andalsopresentseveralexplicitME-LRCsthatareoptimalover
different fields. 2) ME-LRCs with local cyclic codes over F3
Let q = 3 and n′ = 13. We have the following factorization
A. Optimal Construction of x13−1 over F3:
We showhowto construct ME-LRCs which are optimal with x13 1=(x+2)(x3+2x+2)(x3+x2+2)
−
respect to the bound (1) by adding more constraints to Con- (x3+x2+x+2)(x3+2x2+2x+2)
struction A. To this end, we specify the choice of the matrices
We choose the matrices B , B , and B in Construction A as
in Construction A. This specification, referred to as Design I, 1 2 3
follows.Let B betheparity-checkmatrixofa[13,10,3] code
is as follows. 1 3
with the generator polynomial:
1) H1′ is the parity-check matrix of an [n′,n′−v1,d′1]q code C1
which satisfies ko(qp)t[n′,d′1]=n′−v1. g1 = x3+x2+x+2,
B2 be the parity-check matrix of a [13,6,6]3 code 2 with the All of the above three families of ME-LRCs over F4 are
C
generator polynomial: optimal with respect to the bound (1). (cid:3)
g2 =(x+2)(x3+x2+2)(x3+x2+x+2), V. CONCLUSION
In this work, we studied bounds and constructions for ME-
and B be the parity-check matrix of a [13,3,9] code with
3 3 3
C LRCs. We presented a general construction that yields optimal
the generator polynomial:
ME-LRCs with a wide range of parameters when d 6 2d0. A
g3 =(x+2)(x3+x2+2)(x3+x2+x+2) corresponding erasure decoder was also proposed and shown to
(x3+2x2+2x+2). correct any d 1 erasures. Finally, several families of explicit
−
ME-LRCs were constructed over different small fields.
It is clear that .
3 2 1
C ⊂C ⊂C
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0 0
−
APPENDIXA For the last level, i.e., theµth level, if sµ=0, then wq(x)>
̸
PROOFOFLEMMA3 wqvµ(sµ)dµ′ 1 >δµdµ′ 1 >dm, which contradicts our assump-
Proof: We can construct a (ρ,n0,k;d0,d)q ME-LRC in tion. Now, if−s1 =s2 =−··· =sµ=0, then wq(x)>dµ′ >dm,
which also contradicts our assumption.
two steps, and use the GV bound [17] twice. First, there exists
Thus, our assumption is violated.
a [ρ(n r ),k,d] array code of size ρ (n r ) where
0 0 1 0 0
− G × −
r0 is an integer 06r0 <n0, if it satisfies
APPENDIXC
d∑−2 ρ(n0−r0)−1 (q 1)i <qρ(n0−r0)−k. (5) PROOFOFTHEOREM6
i=0( i ) − Proof:TheME-LRCCA hasd0 =d′1 andd =dµ′ .Assume
Second, there exists a length-n code with minimum dis- thereexistdµ′ −1erasuresandadditionalρ′ rows,eachwithless
tance d , if its redundancy r sati0sfies G2 than or equal to d′1−1 erasures. We show that the decoder DA
0 0 can correct these erasures. The proof proceeds by induction.
r0 >logq d∑0−2 n0−i 1 (q−1)i . (6) torF(osr11t,h·e··fi,rsst1ℓ)levisel,thseinaclel-δze1ro=vec,totrh,ei.ceo.,rr(esc11t,s·y·n·d,ros1ℓm)e=vec0-.
( i=0 ( ) ) Thus, theρ′ rows are corrected. T∞he remaining uncorrected row
byNaodwdi,nwger0enmcoodree reeadcuhnrdoawncoyfstyhmebcooldse. GTh1eusreinsgulttihnegccooddeeGi2s s(bulcohckro)wcˆsj,(jbl∈ocFks,)hiassleastsletahsatndδ′12e,rbaescuareuss.eTifheittoistaδl2nuomr lbaerrgeorf,
a (ρ,n0,k;d0,d)q ME-LRC.Letr0 =⌈logq ∑id=0−02(n0i−1)(q− theTnhwuse,hfaovreδth2ed′1se6codnµ′d−le1v,ewl,hitchhe vcioorlaretecst dsµ′yn6drδo2mde′1.vector
1)i , and substitute it into (5), producing (3).
⌉ ( (s2, ,s2) can be obtained. Similarly, the left uncorrected
1 ··· ℓ
) APPENDIXB roofwsucˆcjh, rjow∈sFis,lheasss tahtalneaδst.d′2 erasures, and the total number
3
PROOFOFTHEOREM4 By induction, if the decoder runs until the µth level, then
over FPqro,odfe:nAotceoddbewyoxrd=x(inx1C,µGxT2,P.i.s.,axnℓn),′ℓw-dhiemreenxsiioinnalxveisctaonr aalrletkhneocwonrr.eTcthesryendmroumstebveeoctnolryso(nsei1,u·n·c·o,rsreiℓc),teid=row1,2le,f.t.w.,iµth,
n′-dimensional vector, for i =1,2,...,ℓ. number of erasures at least dµ′ 1 and at most dµ′ −1. Thus, it
Letsj = x H′T,fori =1,2,...,ℓand j =1,2,...,µ.Thus, can be corrected. −
i i j
j
si is a vj-dimensional vector over Fq, and is considered as an
element in Fqvj. Let sj = (s1j,s2j,...,sℓj), an ℓ-dimensional
vector over Fqvj, whose components are sij, i =1,2,...,ℓ.
To prove Theorem 4, we need to show that if xHT = 0 and
wq(x) < dm = min{δ1,δ2d′1,δ3d′2,··· ,δµdµ′ 1,dµ′ }, then x
must be the all-zero vector 0. −
Weproveitbycontradictionandinduction.Assumethatthere
exists a codeword x such that xHT = 0, w (x) < d , and
q m
x =0.
̸
Wefirststateapropositionwhichwillbeusedinthefollowing
proof.
Proposition8.IfxHT = 0ands1 = s2 = = sj = 0,then
···
wq(xi)>d′jforxi ̸=0,i =1,2,...,ℓ.
Proof: The condition s1 = s2 = = sj = 0 means
···
that x BT = 0 for i = 1,2,...,ℓ; that is, x is a codeword in
i j i
a code defined by the parity-check matrix B , whose minimum
j
distance is d′j. Therefore, we have wq(xi) > d′j for xi ̸= 0,
i =1,2,...,ℓ.
Now, if s1 ̸= 0, then wq(x) > wqv1(s1) >δ1 > dm, which
contradicts the assumption. Thus, we have s1 =0.
(a)
Then, consider the second level. If s2 = 0, then wq(x) >
̸
wqv2(s2)d′1 > δ2d′1 > dm, where step (a) is from Proposition
8. Thus, it contradicts the assumption, so we have s2 = 0. By
induction, we must have s1 =s2 = =sµ 1 =0.
−
···