Codes with Unequal Locality

0.17 MB·

by  Swanand Kadhe

Codes with Unequal Locality
Swanand Kadheand AlexSprintson
ElectricalandComputerEngineering
TexasA&MUniversity
[email protected], [email protected]
6 For a code C, its i-th symbol is said to have locality r if its value can be recovered by accessing
1
someotherrsymbolsofC. Locallyrepairablecodes(LRCs)arethefamilyofcodessuchthatevery
0
2 symbolhaslocalityr.
Inthispaper,wefocuson(linear)codeswhoseindividualsymbolscanbepartitionedintosubsets
n
a suchthatsymbolsinonesubsethavedifferentlocalitythantheonesinother. Wecallsuchcodesas
J codeswithunequallocality. Forcodeswithunequalinformationlocality,wecomputeatightupper

2 boundon the minimum distance as a function of number of informationsymbols of each locality.
2 WedemonstratethattheconstructionofPyramidcodescanbeadaptedtodesigncodeswithunequal

informationlocalitythatachievetheminimumdistancebound. Thisresultgeneralizestheclassical
]
T resultof Gopalanet al. for codeswith unequallocality. Next, we consider codeswith unequalall
I symbollocality,andestablishanupperboundontheminimumdistanceasafunctionofnumberof
.
s symbolsofeachlocality.Weshowthattheconstructionbasedonrank-metriccodesbySilbersteinet
c
al.canbeadaptedtoobtaincodeswithunequalallsymbollocalitythatachievetheminimumdistance
[
bound. Finally, we introduce the conceptof locality requirement on a code, which can be viewed

1 asarecoverabilityrequirementonsymbols. Informationlocalityrequirementonacodeessentially
v
specifies the minimum number of information symbols of different localities that must be present
3
in the code. We presenta greedyalgorithm that assigns localities to informationsymbols so as to
5
1 maximizetheminimumdistanceamongallcodesthatsatisfyagivenlocalityrequirement.
6
0
.
1 1 Introduction
0
6
1 Coding for distributed storage has recently attracted significant research attention with a focus on the
v: problem of recovery from storage node failures. The thrust has been on characterizing fundamental
i limitsanddesigning associated codingschemesforoneormoreofthefollowingmetricsthatarecrucial
X
in the node repair process: (a) repair bandwidth – the amount of data downloaded during failed node
r
a repair[1,2];(b)diskI/O–thenumberofbitsreadfromthenodesparticipatingintherepairprocess[3,4];
and(c)repairlocality–thenumberofnodesparticipating intherepairprocess[5,6].
Inthispaper,wefocusonthemetricofrepairlocalityandaclassofcodesdesignedinthecontextof
this metric, known as locally repairable codes (LRCs). Consider a block code of length n that encodes
k information symbols. Asymbol iissaidtohave locality r ifitcanberecovered byaccessing r other
i i
symbols in the code. Wesay that a code has information locality r if each of its k information symbols
haslocality atmostr. Similarly,wesaythatacodehasall-symbol locality rifeachofitsnsymbolshas
locality atmostr.
Codeswithsmalllocality wereintroduced in[7,8](seealso [6]). Thestudy ofthelocality property
was galvanized with the pioneering work of Gopalan et al. [5]. One of their key contributions was to
establish a trade-off between the minimum distance of a code and its information locality analogous
to the classical Singleton bound. In particular, the authors showed that for a (scalar) linear (n,k) code
Submittedto:
ArXivPre-print
2 CodeswithUnequalLocality
having information localityr,theminimumdistance d ofthecodeisupperbounded as
k
d≤n−k− +2. (1)
r
(cid:24) (cid:25)
They also demonstrated that the Pyramid code construction in [7] achieves this bound. Since then, a
series of results have extended the code distance bound for a given locality for various types of codes
along withcorresponding optimalcodeconstructions achieving thedistance bound. Wegiveabrief(far
fromcomplete)overviewofsomeoftheseresultsbelow.
Related work: The distance bound was generalized for codes with multiple local parities in [9],
universal(scalar/vectorlinear,nonlinear)codesin[10],universalcodeswithmultipleparitiesin[11,12].
Aninteger programming based bound wasestablished in[13]. Almostalloftheseworksalsopresented
optimalcodeconstructions. Furthermore, alarge numberofotheroptimalcodeconstructions havebeen
presented, see e.g., [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The notion of locality was extended to
multiple recovery groups (also known as, availability) in [24, 25], and for the case of multiple failures,
to sequential repair in [26] and hierarchical repair in [27]. The Singleton-like bound was extended to
accommodate thealphabetsizein[28].
Ourcontributions: Inprevious works, thelocality ofacode ischaracterized byasingle parameter
r. Inspired from the notion of unequal error protection, we are interested in investigating linear codes,
in which, different subsets ofsymbols possess different localities. Werefer tosuch codes as codes with
unequal locality. Forexample, consider a(15,11) codewhose4information symbolshave locality 2,3
information symbols have locality 3, and 4 information symbols have locality 4 (with no constraint on
the locality of parity symbols). Under the classical terminology, such a code would be characterized as
acode withinformation locality 4. However,itisnotclear ifthedistance bound givenin(1)istight for
thecaseofunequallocalities. Ourmaingoalistocomputeatightupperboundontheminimumdistance
ofsuchcodeswithunequal locality.
Codeswithunequal localityarepractically appealing inscenarios whenimportantinformation sym-
bols, e.g.,symbolsofhotdata, needtoberepaired quickly; whereas, recovering lessimportant symbols
can involve moreoverhead. Moreover, these types ofcodes can beuseful inreducing download latency
forhotdata. Forinstance, references[29,30]studystoragecodesfromqueueingtheoreticperspectiveto
analyze downloadlatency.
Our key contributions are summarized as follows. To characterize a code with unequal information
locality,wedefineanotionofinformationlocalityprofileofacode. Wesaythatacodehasaninformation
locality profile k={k ,...,k } if it contains k information symbols of locality j for 1 ≤ j ≤ r. For
1 r j
example, a code having 5 information symbols of locality 2, and 6 information symbols of locality 4
would have an information locality profile {0,5,0,6}. For scalar linear codes, we establish an upper
bound ontheminimum distance asafunction ofinformation locality profilek={k ,...,k }asfollows
1 r
(Theorem1).
(cid:229) r kj
d≤n−k− +2. (2)
j
j=1(cid:24) (cid:25)
We demonstrate that the Pyramid code construction in [7] can be adapted to design unequal locality
codesthataredistance-wise optimalaccording totheboundabove.
Whenparitysymbolsalsohavelocalityconstraints, wecananalogouslydefineanallsymbollocality
profile of a code. W say that a code has an all symbol locality profile n={n ,...,n } if it contains k
1 r j
informationsymbolsoflocality jfor1≤ j≤r. Forinstance,considera(15,11)codethathas6symbols
oflocality2,4symbolsoflocality3,and5symbolsoflocality4. Itsallsymbollocalityprofilewouldbe
S.KadheandA.Sprintson 3
{0,6,4,5}. We compute an upper bound on the minimum distance for scalar linear codes as a function
ofallsymbollocalityprofile,whichhasthefollowingform(Theorem2).1
r(cid:229)−1 nj k−(cid:229) ri=−11 nj− jn+j1
d ≤n−k+2− − . (3)
j+1  (cid:16)r l m(cid:17)
j=1(cid:24) (cid:25)
 
We adapt the construction in [14], which uses a maximum rank distance (MRD) code as an outer code
 
andamaximumdistanceseparable(MDS)codeasinnercode,toconstructcodeswithunequalallsymbol
locality thatareoptimalwithrespect totheabovebound.
Finally, we introduce a concept of information locality requirement. To motivate this, consider a
scenariowhereweneedtodesignalinearcodeofdimensionk=11suchthatk˜ =5informationsymbols
3
musthavelocalityatmost3,andtheremainingk˜ =6informationsymbolsmusthavelocalityatmost4.
4
Collectively,wecanspecifythisasalocalityrequirementofk˜ ={0,0,5,6}. Noticethatthisisequivalent
toarequirement asacodemustcontainatleast5symbolsoflocality upto3,andatleast11symbolsof
locality up to 4. In general, alocality requirement of k˜ ={k˜ ,...,k˜ } means that acode should contain
1 r
atleast(cid:229) i k˜ symbolsoflocality uptoiforeach1≤i≤r,or,inotherwords, k˜ information symbols
i=1 j j
shouldhavelocalityatmost j.
One can design codes with various information locality profiles that would satisfy this require-
ment. For examples, the locality requirement of k˜ ={0,0,5,6} is satisfied by locality profiles {5,6},
{0,0,5,6}, {0,2,9}, {1,0,6,4}, etc. The question what is the maximum value of minimum distance
any code with this locality requirement would attain, and can we find an optimal locality profile which
achieves thisdistance? Notethatlocality requirement canbeviewedasarecoverability requirement for
code design. We give a simple greedy algorithm which computes an information locality profile given
aninformation localityrequirement.
2 Preliminaries
2.1 Notation
Weusethefollowingnotation.
1. Foranintegerl,[l]={1,2,...,l};
2. For a vector x and an integer i, x(i) denotes the i-th coordinate of x, for a matrix H and integers
i,j,H(i,j)denotestheelementinrowiandcolumn j;
3. Foravectorx,Supp(x)denotesitssupport, i.e.,Supp(x)={i:x(i)6=0};
4. Foravectorx,wt(x)denotes itsHammingweight,i.e.,wt(x)=|Supp(x)|;
5. Forvectorsxandy,x·ydenotes theirdotproduct;
6. Forasetofvectorsx ,...,x ,hx ,...,x idenotestheirspan,whereasforamatrixH,hHidenotes
1 m 1 m
itsrowspace;
7. ForavectorspaceA,dim(A)denotesitsdimension;
8. ForamatrixH,rank(H)denotes therankofH.
1Inanparallelandindependentwork,ZehandYaakobi[31]alsoconsidertheproblemofcomputingaboundonminimum
distanceofcodeswithunequalallsymbollocality,referredintheirworkasmultiplelocalitycodes.Theirbound[31,Theorem
8]hasasimilarformasweget.Inaddition,[31]extendsCadambe-Mazumdarboundin[28]forcodeswithmultiplelocalities,
andpresentseveraloptimalcodeconstructions.
4 CodeswithUnequalLocality
2.2 Codes with Locality
Let C denote a linear [n,k,d] code over F with block-length n, dimension k, and minimum distance
q q
d. Let c denote a codeword in C. The code can be represented by a set of n (column) vectors C =
{c ,...,c }∈Fk. The set of vectors must have rank k for C to have dimension k. The i-th vector c is
1 n q i
referred toasthei-thcoordinate ofC. Foranycodeword c∈C,c(i)issaid tobethei-thsymbolofthe
codewordc. Inthecontextoflocality,weusethetermssymbolorcoordinateinterchangeably. Ourmain
focus is on systematic codes, and we assume that the first k coordinates correspond to the information
symbols.
We say that the i-th coordinate of a code C has locality r if its value can be recovered from some
i
otherr coordinates ofC. Theformaldefinitionoflocality isasfollows.
i
Definition1. [Locality] Forc ∈C,wedefineLoc(c)tobethesmallestinteger r suchthatthereexists
i i i
asubsetR(i)⊂[n]\{i},|R(i)|≤r,suchthatc =(cid:229) l c ,wherel ∈F ∀l∈R(i).
i i l∈R(i) l l l q
Note that, if the minimum distance of the code is more than two, then every coordinate has locality
atmostk.
Wesaythatan(n,k)codehasinformationlocalityrifeachofitskinformationsymbolshaslocality
at most r. Similarly, we say that an (n,k) code has all symbol locality r if each of its n symbols has
locality atmostr.
3 Codes with Unequal Information Locality
Inthissection,weareinterestedinsystematiccodes,whoseinformationsymbolscanbepartitionedinto
disjoint subsets in such a way that the symbols in one subset have different locality than the symbols
in other subset. We say that such codes possess unequal information locality. We can characterize the
locality of such codes by listing the locality values of each information symbol. Alternatively, we can
consider thelistofcardinalities ofsubset ofeach locality. Wecall suchalistastheinformation locality
profileofthecode. Formally,thedefinition isasfollows.
Definition2. [Information Locality Profile]Givenasystematic [n,k,d] codeC,theinformation local-
q
ity profile of C is defined as a length-k vector r(C)= {r ,...,r }, where r is the locality of the i-th
1 k i
information coordinate ofC. Notethat1≤r ≤kforeachi∈[k],assumingd ≥2.
i
Alternatively, wecanspecifythelocalityprofileofC asalength-r vectork(C)={k ,...,k },where
1 r
r=max{r ,...,r } and k is the number of information coordinates of locality j for j∈[r]. Note that
1 k j
∀j∈[r],0≤k ≤k,k ≥1and(cid:229) r k =k.
j r j=1 j
Remark1. ForacodeC withrepresentationC,wecanchoose anysubset ofk full-rank coordinates of
C to represent information symbols. Without loss of generality, we can always choose the coordinates
having smallest overall locality asinformation coordinates. Morespecifically, for1≤ j≤r, letC ⊂C
j
bethesubsetofcoordinates havinglocality j. SetC =0/. Let
0
k =rank ∪j C −rank ∪j−1C . (4)
j i=0 i i=0 i
(cid:16) (cid:17) (cid:16) (cid:17)
In other words, (cid:229) j k is the rank of the sub-matrix formed by the coordinates having locality up to j.
i=1 i
Starting with j =1, we choose a subset I ⊂C of k linearly independent coordinates to represent k
j j j j
information symbols, andcontinue incrementing jtillthetotalrankisk.
S.KadheandA.Sprintson 5
Algorithm1ConstructsetS⊆C suchthatrank(S)≤k−1
1: LetS0=0/,i=1
2: whilerank(Si−1)≤k−2do
3: Pickacoordinate ci∈C\Si havingsmallestlocality
4: ifrank Si−1∪cG (i) <kthen
5: SetS(cid:0)i=Si−1∪cG(cid:1)(i)
6: else
7: PickG (i)′⊂G (i)suchthatrank Si−1∪cG (i)′ =k−1
8: SetSi=Si−1∪cG (i)′ (cid:16) (cid:17)
9: endif
10: Incrementi
11: endwhile
Remark 2. In the classical notion of locality defined by Gopalan et al. [5], technically, every symbol
can have different locality. However, the (information) locality of a code is parameterized by a single
value r, which is the largest locality of an (information) symbol. On the other hand, we parameterize
theinformation locality usingalength-k vectorthatspecifies thelocality ofeachindividual information
symbol. Weareinterested incharacterizing atrade-off between the minimum distance ofacode and its
locality profilevector.
3.1 Bound onthe Minimum Distance
Consider a class of systematic linear codes having an information locality profile k={k ,...,k }. We
1 r
are interested in finding an upper bound on the minimum distance as a function of the code length,
dimension, andinformation locality profile. Thiswouldbeageneralization oftheresult in[5]forcodes
withunequallocalities forinformation symbols.
Theorem 1. For any linear code with block-length n, dimension k, and information locality profile
k={k ,...,k },wehave
1 r
(cid:229) r kj
d≤n−k− +2. (5)
j
j=1(cid:24) (cid:25)
Proof. Webuild on the proof technique proposed in[5]. The idea isto construct alarge set S⊆C such
thatrank(S)≤k−1,andthenusethefollowingfact.
Fact1. ([5])ThecodeC hasminimumdistancedifandonlyifforeveryS⊆Csuchthatrank(S)≤k−1,
wehave
|S|≤n−d. (6)
RecallthatR(i)denotesarepairgroupofc,andwehave|R(i)|=Loc(c). DefineG (i):={i∪R(i)}.
i i
Further, foranysubsetT ⊆[n],definec ={c ∈C:i∈T}.
T i
We use Algorithm 1 to construct a set S such that rank(S) < k. First, note that in line 3, as
rank(S )≤k−2,andtherearek(linearlyindependent)informationsymbols,thereexistsacoordinate
i−1
c ∈/ S .
i i−1
Our goal is to find a lower bound on |S|. Let l be the total number of iterations of Algorithm 1.
Observethat|S|=|S |. Further,thefinalsetS hasrank(S )=k−1. Wedefinetheincrementinthesize
l l l
6 CodeswithUnequalLocality
andrankofsetS inthei-thiterationasfollows.
i
s =|S|−|S |, t =rank(S)−rank(S ). (7)
i i i−1 i i i−1
Notethat
l l
(cid:229) (cid:229)
|S |= s, rank(S )= t =k−1. (8)
l i l i
i=1 i=1
We consider two cases depending on whether Algorithm 1 reaches the condition in line 4, i.e.,
rank Si−1∪cG (i) =k. Wenotethatthecondition canbereached onlyinthelastiteration.
C(cid:0)ase 1: Supp(cid:1)ose we have rank Si−1∪cG (i) ≤k−1throughout. Now, in the i-th iteration, weadd
cG (i) toS. Thus,si≤Loc(ci)+1. Fu(cid:0)rther,vector(cid:1)sincG (i)\Si−1 aresuchthattheyyielda(possiblyzero)
vectorinhS i. Therefore,
i−1
t ≤s −1≤Loc(c). (9)
i i i
Usingthis,wecanwrite
l l
(cid:229) (cid:229)
|S|= s ≥ (t +1)=k−1+l, (10)
i i
i=1 i=1
wherethelastequalityfollowsfrom(8).
Lowerbounding thenumberofiterations. Now,tofindalowerbound on|S|,wefindalowerbound
on l. Let m be the locality of the last symbol collected by Algorithm 1, where m∈[r]. For 1≤ j≤m,
let l be the number of iterations in which Algorithm 1 picks coordinates of locality j. Note that, if C
j
does not contain any symbol of a particular locality j, we set l =0. Thus, for each j, 0≤l ≤l, and
j j
l=(cid:229) m l .
j=1 j
RecallthatC ⊂Cisthesetofcoordinates oflocality j(seeRemark1). Sincethealgorithmcollects
j
allcoordinatesoflocalityupto jbeforecollectinganycoordinateoflocality j+1for1≤ j≤m−1,we
have S =∪j C . Therefore, from (17), rank(S )=k and for 2≤ j≤m−1, rank S −
(cid:229) pj=1lp p=1 p l1 1 (cid:229) pj=1lp
(cid:16) (cid:17)
rank S =k . Thisresultsin
(cid:229) pj−=11lp j
(cid:16) (cid:17)
j
(cid:229)
rank S = k , for 1≤ j≤m−1. (11)
(cid:229) pj=1lp p=1 p
(cid:16) (cid:17)
Theabovetworesults canbeinterpreted asfollows. Theincrement intherankofSbycollecting allthe
coordinates of locality j is k for 1≤ j≤m−1. The rank of S, when it contains all the coordinates of
j
locality upto j,is(cid:229) j k .
p=1 p
Whenthealgorithm terminates,itmaynothavecollectedallthecoordinates oflocalitym. Letk′ be
m
the increment inthe rank ofSby thecoordinates oflocality mthat are collected by thealgorithm. Note
that1≤k′ ≤k .
m m
Note that rank(S ) = rank S +k′ . Using the fact that rank(S ) = k−1 and (11), we get
l (cid:229) mj=−11lj m l
k−1=(cid:229) m−1k +k′ . On the o(cid:16)ther han(cid:17)d, by definition of locality profile vector, we have (cid:229) r k =k.
j=1 j m j=1 j
Weconsider twocases.
Case(1a): k ≥21. Then,itmustbethatm=randk′ =k −1since1≤k′ ≤k .
r m r m m
Case(1b): k =1. Then,itfollowsthatm=r−1,andk′ =k since1≤k′ ≤k .
r m r−1 m m
Insummary,for1≤ j≤r−1,theincrementintherankofSbycollectingthecoordinatesoflocality
j isk . Theincrement in therank ofS by locality r coordinates isk −1. (Note that this holds for Case
j r
S.KadheandA.Sprintson 7
(b) as well.) Moreover, for each 1≤ j≤r, when the algorithm is collecting the coordinates of locality
j,therankcanincreasebyatmost j ineachstep(see(9)). Therefore, l ≥ kr−1 for1≤ j≤r−1and
j r
l ≥ kr−1 . l m
r r
Clombimningthiswithl=(cid:229) rj=1lj gives,
r(cid:229)−1 kj kr−1
l≥ + . (12)
j r
j=1(cid:24) (cid:25) (cid:24) (cid:25)
Substituting thisinto(10),weget
r(cid:229)−1 kj kr−1
|S|≥k−1+ + (13)
j r
j=1(cid:24) (cid:25) (cid:24) (cid:25)
(cid:229) r kj
≥k−2+ . (14)
j
j=1(cid:24) (cid:25)
Case 2: In the last step, we get rank Sl−1∪cG (l) =k. For 1≤i≤l−1, in the i-th iteration, we
addcG (i). Thus,si≤Loc(ci)+1. Further,(cid:0)vectorsincG(cid:1)(i)\Si−1 aresuchthattheyyielda(possibly zero)
vector in hS i. Therefore, for 1≤i≤l−1, we get t ≤s −1≤Loc(c). In the last step l, we add
i−1 i i i
cG (l)′ ⊂cG (l). Thisincrementsrank(S)bytl ≥1(sincerank(Sl−1)≤k−2),and|S|bysl ≥tl. Therefore,
wehave
l l−1
(cid:229) (cid:229)
|S|= s ≥ (t +1)+t =k−1+l−1, (15)
i i l
i=1 i=1
thelastequality followsfrom(8).
Lower bounding the number of iterations. Similar to Case 1, in each iteration i (including the last
one),wehavet ≤Loc(c). Theonlydifference fromCase1isthatSaccumulatestotalrankofkinstead
i i
ofk−1. Therefore,tolowerboundl,wecanusethesameargumentsasinCase1alongwithrank(S )=k
l
toobtainl≥(cid:229) r−1 kj + kr inplaceof(12). Substituting thisinto(15)yields|S|≥k−2+(cid:229) r kj
j=1 j r j=1 j
(whichissameas(l14)m). l m l m
Finally, notingthat|S|≤n−d fromFact1andusingthislowerboundon|S|gives(5).
3.2 Code Construction: PyramidCodes
We show that the parity splitting construction of the Pyramid codes [7] can be adapted to obtain codes
withunequal information locality, thatare optimal withrespect to(5). Consider aninformation locality
profile k={k ,...,k }. Let {j ,...,j } with j <···< j be the m(≤r) localities such that k >0.
1 r 1 m 1 m jp
We begin with a (k+d−1,k,d) systematic maximum distance separable (MDS) code C′. Let the
representingcoordinatesbeC′ ={e ,...,e ,p ,...,p },wheree isthe j-thcolumnofak×kidentity
1 k 0 d−2 j
matrix,andp for0≤ j≤d−2arethecolumnsrepresenting theparitycoordinates.
j
Wepartition theset[k]intomdisjoint subsets S ,...,S such that|S |=k foreach p∈[m]. Next,
1 m p jp
partition each subset S into l = kjp disjoint subsets each of size at most j . That is, S =∪lp S .
p p jp p p i=1 p,i
Foravectorxofdimension k,andlasemtS⊆[k], letx|S denote the |S|-dimensional restriction ofxtothe
coordinates insetS. Then,wedefinethesystematic codeC withthefollowingrepresentation.
C= e ,...,e ,p | ,...,p | ,p | ,...,p | ,...,p | ,...,p | ,p ,...,p . (16)
1 k 0 S1,1 0 S1,l1 0 S2,1 0 S2,l2 0 Sm,1 0 Sm,lm 1 d−2
n o
8 CodeswithUnequalLocality
Note that we have split the parity p into (cid:229) r kj parities. Therefore, n=k+d−2+(cid:229) r kj .
0 j=1 j j=1 j
It is easy to verify that parity splitting does not affelct mthe distance, and hence, the code C has disltancme
d. Since C′ is an MDS code, we have wt(p )=k. Therefore, a set of k information coordinates and
0 jp
kjp paritycoordinates havelocalityatmost j foreach p∈[m]. SimilartotheclassicalPyramidcodes
jp p
lin[7m],thelastd−2paritysymbolsmayhavelocality aslargeask.
4 Codes with Unequal All Symbol Locality
Inthis section, weextend thenotion ofinformation locality toprofiletoaccommodate thecodes whose
parity symbols also have locality constraints. Inthis case, code symbols can bepartitioned into disjoint
subsets according to their locality, with maximum locality r <k. We define all symbol locality profile
a
ofacodeasfollows.
Definition 3. [All Symbol Locality Profile] Given an [n,k,d] code C, the all symbol locality profile of
q
C isdefinedasalength-n vectorr(C)={r ,...,r },wherer isthelocalityofthei-thcoordinate ofC.
1 n i
Notethat1≤r ≤kforeachi∈[n],assumingd ≥2.
i
Alternatively, we can specify the locality profile of C as a length-r vector n(C) = {n ,...,n },
a 1 ra
where r =max{r ,...,r } and n is the number of information coordinates of locality j for j ∈[r ].
a 1 n j a
Notethat∀j∈[r ],0≤n ≤n,n ≥1and(cid:229) ra n =n.
a j ra j=1 j
Remark3. Foracode C withrepresentationC,letC ⊂C bethesubset ofcoordinates having locality
j
jfor1≤ j≤r . Ifn =0forsome j,thenwesetC =0/. For1≤ j≤r ,wedefine
a j j a
k =rank ∪j C −rank ∪j−1C , (17)
j i=0 i i=0 i
(cid:16) (cid:17) (cid:16) (cid:17)
wherewesetC =0/. Definer=max{j:k >0}. Then,{k ,...,k }canbeconsideredastheinformation
0 j 1 r
locality profile of C. Codes with the same all symbol locality profile can have different information
locality profiles.
4.1 Bound onthe Minimum Distance
Notethatcodeswithunequallocalitiesforallsymbolsareaspecialclassofcodeswithunequalinforma-
tionlocalities. Therefore,theminimumdistanceupperboundin(5)holdsforanallsymbollocalitycode
having information locality profile k. As noted in Remark 3, it is possible for a code to have different
information locality profiles for a given all symbol locality profile. The upper bound in (5) obtained
using only information locality profile maynotbetight forcertain information localities. Ourgoalisto
computeanupperboundontheminimumdistance asafunction ofallsymbollocalityprofile.
Theorem2. ConsideracodeC withallsymbollocalityprofilen={n ,...,n }. Definek′ =n − nj .
1 ra j j j+1
Letr′=max{1≤i≤r :(cid:229) i k′ <k}. Letr=min{r′+1≤ j≤r :n ≥2}. Then,wehave l m
a j=1 j a j
r(cid:229)−1 nj k−(cid:229) ri=−11 nj− jn+j1
d ≤n−k+2− − . (18)
j+1  (cid:16)r l m(cid:17)
j=1(cid:24) (cid:25)
 
Proof. Similar to information locality case, we consider Algorithm 1 to find a set S ⊂C such that
 
rank(S)≤k−1.
S.KadheandA.Sprintson 9
RecallthatC ⊂C isasubsetofcoordinates oflocality j. Letk =rank ∪j C −rank ∪j−1C ,
j j i=0 i i=0 i
wherewedefineC0=0/. (cid:16) (cid:17) (cid:16) (cid:17)
It is easy to show that k ≤ k′ for each 1 ≤ j ≤ r. In particular, consider the following greedy
j j
algorithm. BeginningwithT =0/ untilT =Cj,ineachiteration p,extendT asbyaddingacoordinate
0 p p−1
cp∈Cj\Tp−1 andallitsrepairgroupcoordinates cR(p) toTp−1. Specifically, Tp=Tp−1∪(cG (p)\Tp−1).
Now, in each iteration there must be at least one linear dependency between Tp−1 and cG (p)\Tp−1.
Further, ineachiteration, weextend thesize ofT byatmost j, andthus, thenumber ofiterations areat
least nj . Therefore, thenumberoflinear dependencies amongthecoordinates inC mustbeatleast
j+1 j
nj l. m
j+1
l Cmase 1: Suppose we have rank Si−1∪cG (i) ≤k−1 throughout. Let m be the locality of the last
symbolpickedbythealgorithm. For1≤ j≤m−1,thealgorithm collectsallthecoordinates oflocality
(cid:0) (cid:1)
j. Letnˆ ≤n bethenumberofcoordinates oflocality mthatarecollected bythealgorithm. Then,we
m m
have
|S|=n +···+n +nˆ .
1 m−1 m
Notethatrank(S)whenShasaccumulatedallthecoordinatesoflocalityuptom−1isrank ∪m−1C =
j=1 j
(cid:229) m−1k . Therefore, therankaccumulated from locality mcoordinates is(k−1)−(cid:229) m−1k (cid:16):=kˆ . N(cid:17)ow,
j=1 j j=1 j m
using standard arguments similar to the proof of Theorem 1, it is easy to show that nˆ ≥ kˆ + kˆm .
m m m
Therefore, l m
|S|≥m(cid:229) −1n +kˆ + kˆm :=|S| . (19)
j m LB
m
j=1 & '
Next, weshow that |S| is minimized when k =k′. Let S′ be the set collected ifrank ∪j C −
LB j j i=0 i
rank ∪j−1C =k′. In this case the locality of the last coordinate must be r provided (cid:229) r−1(cid:16)k′ <k−(cid:17)1.
i=0 i j j=1 j
Letnˆ(cid:16)′ bethe(cid:17)numberofcoordinates oflocality rthatarecollected bythealgorithm. (If(cid:229) r−1k′ =k−1,
r j=1 j
thennˆ′ =0andthefollowinganalysisstillholds.) Then,wehave
r
′
|S|=n +...+n +nˆ .
1 r−1 r
The rank accumulated in locality r coordinates is (k−1)−(cid:229) r−1k′ :=kˆ′. Again, using standard argu-
j=1 j r
mentssimilartotheproofofTheorem1,itiseasytoshowthatnˆ′ ≥kˆ′+ kˆr′ . Therefore,
r r r
l m
|S′|≥r(cid:229)−1n +kˆ′+ kˆr′ :=|S′| . (20)
j r r LB
j=1 & '
Next, we show that |S′| ≤|S| . Suppose, for contradiction, |S′| >|S| . First, note that since
LB LB LB LB
k′ ≥kjjfor1≤ j≤r,wehaver≤m.
j
Case(1a): m=r. Then,wehave
r(cid:229)−1n +kˆ′+ kˆr′ >r(cid:229)−1n +kˆ + kˆr .
j r r j r r
j=1 & ' j=1 & '
However,thisessentially implies(cid:229) r−1k′ <(cid:229) r−1k ,whichisacontradiction.
j=1 j j=1 j
10 CodeswithUnequalLocality
Case(1b): m<r. Then,wehave
r(cid:229)−1n +kˆ′+ kˆr′ >r(cid:229)−1n +n +···+kˆ + kˆr .
j r r j r r r
j=1 & ' j=1 & '
However,thisimplieskˆ′+ kˆr′ >n +···+kˆ + kˆr ,whichisacontradiction askˆ′+ kˆr′ ≤nˆ′ ≤n .
r r r r r r r r r
Hence, to get smallestllowmer bound on |S|, olnemcan assign maximum incrementallranmk k′ to each
j
locality j. Letl bethenumberofiterations duringwhichAlgorithm1collectscoordinates oflocality j.
j
Then, using the same arguments as inthe proof of Theorem 1, wehave |S|≥k−1+(cid:229) r l (see (10)).
j=1 j
For 1 ≤ j ≤ r−1, the algorithm collects all the n coordinates of locality j. When a coordinate of
j
locality j is picked, the size of S can be increased by at most j+1 in that iteration. Thus, l ≥ nj
j j+1
for 1≤ j≤r−1. Forlocality r, we increment the rank of S by (k−1)−(cid:229) r−1k′. Ateach step, tlank ims
j=1 j
(k−1)−(cid:229) r−1k′
increased byatmostr,thusl ≥ j=1 j . Hence,
r r
(cid:24) (cid:25)
r(cid:229)−1 nj (k−1)−(cid:229) rj−=11k′j r(cid:229)−1 nj k−(cid:229) rj−=11k′j
|S|≥k−1+ + ≥k−2+ + .
j+1 r j+1 r
j=1(cid:24) (cid:25) & ' j=1(cid:24) (cid:25) & '
Case 2: In the last step, we get rank Sl−1∪cG (l) = k. Analysis to show that the smallest lower
boundon|S|isobtainedassigning maximumincremental rankk′ toeachlocality jissimilartoCase1.
(cid:0) (cid:1) j
Using the same arguments as in the proof of Theorem 1, we have |S|≥k−2+(cid:229) r l (see (15)).
j=1 j
Following the same argument asCase 1, l ≥ nj for 1≤ j≤r−1. Forlocality r, weincrement the
j j+1
rankofSbyk−(cid:229) r−1k′. Ateachstep,tankislincreamsed byatmostr,thusl ≥ k−(cid:229) rj−=11k′j . Hence,
j=1 j r r
(cid:24) (cid:25)
r(cid:229)−1 nj k−(cid:229) rj−=11k′j
|S|≥k−2+ + .
j+1 r
j=1(cid:24) (cid:25) & '
Finally,theresultfollowsfromusingtheFact1.
4.2 Code Construction
We adapt the rank-metric codes based LRC construction in [14] for the unequal all symbol locality
scenario. Theideaistofirstprecodetheinformationsymbolswitharank-metriccode(inparticular,with
Gabidulin codes), and then use maximum distance separable (MDS) codes to obtain local parities. We
beginwithabriefreviewofrank-metric codes.
4.3 Rank-Metric Codes
Let FN×m be the set of all N×m matrices over F . The rank distance is a distance measure between
q q
elements AandBofFN×m definedasd (A,B)=rank(A−B). Itcanbeshownthattherankdistance is
q R
indeed a metric [32]. A rank-metric code is a non-empty subset of FN×m under the context of the rank
q
metric.
Typically,therank-metriccodesareconsideredbyleveragingthecorrespondence betweenF1×mand
q
anextension fieldFqm. ByfixingabasisforFqm asanm-dimensional vectorspaceoverFq,anyelement