Error Correcting Index Codes and Matroids

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by  Anoop Thomas

Error Correcting Index Codes and Matroids
Anoop Thomas and B. Sundar Rajan
Dept. of ECE, IISc, Bangalore 560012, India, Email: {anoopt,bsrajan}@ece.iisc.ernet.in.
Abstract—The connection between index coding and matroid {1,2,...,n}. Each R possesses a set of messages {x } ,
theory have been well studied in the recent past. El Rouayheb where χ ⊆(cid:100)n(cid:99). Let χi ={χ ,χ ,...,χ } representsjthje∈χseit
i 1 2 m
et al. established a connection between multi linear representa- ofsideinformationpossessedatallreceivers.EachreceiverR
i
tion of matroids and wireless index coding. Muralidharan and demandsasinglemessagex .Themappingf :(cid:100)m(cid:99)→(cid:100)n(cid:99)
f(i)
Rajan showed that a vector linear solution to an index coding
satisfiesf(i)∈/ χ .Soanindexcodingproblemisdescribedby
problem exists if and only if there exists a representable discrete i
thequadruple(m,n,χ,f).Theobjectiveistofindanencoding
5 polymatroid satisfying certain conditions. Recently index coding
scheme referred to as an index code that satisfies all receivers
1 witherroneoustransmissionwasconsideredbyDauetal..Error
and uses minimum number of transmissions.
0 correcting index codes in which all receivers are able to correct
2 a fixed number of errors was studied. In this paper we consider Definition 1: AnindexcodeoverF foraninstanceofthe
n amoregeneralscenarioinwhicheachreceiverisabletocorrect indexcodingproblemdescribedby(m,qn,χ,f)isanencoding
a aerrdoersirceodrrencutminbgerinodfexercroodrse,s.caAllinlignksubcehtwinedeenxdcioffdeersendtiifafelreenrrtioarl function C : Fnq → FNq such that for each receiver Ri, i ∈
J correcting index codes and certain matroids is established. We (cid:100)m(cid:99), there exists a decoding function Di : FNq ×Fq|χi| → Fq
1 define matroidal differential error correcting index codes and we satisfying ∀ x ∈ Fn : D (C(x),x ) = x . The parameter
2 show that a scalar linear differential error correcting index code N is called the lengqth ofithe indexχicode. f(i)
exists if and only if it is matroidal differential error correcting

] index code associated with a representable matroid. A linear index code is an index code, for which the
T encoding function C is a linear transformation over F . Such
q
I a code can be described as ∀ x ∈ Fn : C(x) = xL where
s. I. INTRODUCTION L is an n×N matrix over F . Theqmatrix L is called the
c q
[ The index coding problem introduced by Birk and Kol [1] matrix corresponding to the linear index code C. The code C
involves a source which generates a set of messages and a set is referred to as the linear index code based on L.
1
of receivers which demand messages. Each receiver has prior Note that the encoding function C : Fn → FN, can be
v knowledgeofaportionofthemessagecalledside-information, q q
0 viewed as N encoding functions c1,c2,...,cN, where each
06 wofhitchheissidken-oinwfnortmoatthieonsoauvraciel.abTlheeastoaulrlcethuesersectheievekrnsowtolefidngde cFini,sca(xfu)ngcitvioens tfhroemithFnqtra→nsmFiqs.siFoonrothfethseouinrcdeexmceosdsea.geSoxa∈n
5 a transmission scheme of minimum number of transmissions, Nq lenigth index code C can also be described by set of N
0 which satisfies all the demands of the receivers. Bar-Yossef et functions {c ,c ,...,c }, denoted by S(C).
. al. [2] studied the index coding problem and found that the 1 2 N
1
lengthoftheoptimallinearindexcodeisequaltotheminrank Rouayheb et al. [3] proved that the index coding problem
0
of a related graph. has a solution if and only if the equivalent network coding
5
problem is solvable. The equivalent network coding prob-
1
The connectionbetween multi-linear representationof ma-
: lem for the index coding problem is as follows. Consider
v troidsandindexcodingwasstudiedbyElRouayheb,Sprinston an instance of the index coding problem described by the
Xi andGeorghiades[3].ItwasshownbyMuralidharanandRajan quadruple (m,n,χ,f) with an index code C of length N.
[4] that a vector linear solution to an index coding problem
The corresponding equivalent network coding problem for the
ar exists if and only if there exists a representable discrete index coding problem (m,n,χ,f) and index code C is given
polymatroidsatisfyingcertainconditionswhicharedetermined
inFig.1.Theequivalentnetworkcodingproblemhasnsource
by the index coding problem.
nodes each corresponding to the source messages of the index
coding problem. There are two levels of intermediate nodes
Theproblemofindexcodingwitherroneoustransmissions
denoted by w and w for i = 1 to N. Each node w has
was studied by Dau et al. [5]. An index code capable of i i(cid:48) i
n incoming edges, one from each of the source nodes. Each
correcting at most δ-errors at all its receivers is defined as
node w has only one input edge e , from w . The nodes
a δ-error correcting index code. The necessary and sufficient i(cid:48) i i
R ,R ,...,R are the receiver nodes. Each receiver node
conditions for a scalar linear index code to have δ-error 1 2 m
R has two types of input edges, N input edges from each of
correctingcapabilitywasfound.Networklinearnetworkerror- i
the w node and there are |χ | input edges which correspond
correctingcodeswereintroducedearlierbyYeungandCai[6], i(cid:48) i
to the side information possessed by the receiver. The edges
[7]. The link between network error correcting and matroid
corresponding to side information are represented as dashed
theory was established by Prasad and Rajan in [8].
lines in the figure. The side information edges for a receiver
Index coding problems are modeled in the following R are the edges (x ,R ) for j ∈χ . The information flowing
i j i i
way. There is a unique source S having message x = through the edge e of the equivalent network corresponds
i
(x ,x ...,x )∈Fn, where F is a finite field of q elements. to the ith transmission of the index code C. The problem of
1 2 n q q
TherearemreceiversR ,R ,...,R .Let(cid:100)n(cid:99)denotetheset index coding with error correction with receivers capable of
1 2 m
correcting at most δ errors can be transformed to a network x1 x2 xn
error correcting problem. Hence, a natural question that arise
is whether it is possible to link index codes and representable
matroids by first transforming the index coding problem into
equivalent network coding problem as in [3] and then link
error correcting network code to matroids as in [8]. If one
takes this approach then, in the equivalent network, network w1 w2 wN
code should be able to correct mδ errors as each receiver has
δ error correcting capability. Also these errors are restricted e1 e2 eN
to certain edges of the equivalent network. The errors can
occuronlyintheedgesbetweenw andreceivernodes.Among
i(cid:48)
the N edges between wi(cid:48) and Rj, at most δ can be in error. w1′ w2′ wN′
There are m receivers and hence at most mδ errors. Such
case of restriction of errors to a subset of edges was not
considered earlier. We also note that the size of the ground
set of the matroid for which we have to find representation is
smaller compared to the matroid obtained from the equivalent R1 R2 Rm
network coding problem. If we consider mδ errors without
any restriction and equivalent network the size of the ground
Fig.1. Equivalentinstanceofnetworkcodingproblemforanindexcoding
set of the matroid obtained is n + 2|E|, where E is the problem(m,n,χ,f).
edge set of the equivalent network. Number of edges in the
equivalent network for an index coding problem (m,n,χ,f)
(cid:80)m a vector x = (x ,x ,...,x ) ∈ Fn. The support of vector x
is|E|=(n+m+1)N+ |χ |.Thematroidwhichweobtain 1 2 n q
i is defined to be the set supp(x) = {i ∈ (cid:100)n(cid:99) : x (cid:54)= 0}. The
i=1 i
has a ground set of cardinality only n+2N which is much Hamming weight of a vector x, denoted by wt(x), is defined
smaller than n+2|E|. to be the |supp(x)|. For some subset B = {i ,i ,...,i } of
1 2 b
(cid:100)n(cid:99), where i < i < ... < i , let x denote the vector
The contributions of this paper are as follows. 1 2 b B
(x ,x ,...,x ). For some matrix A, A(i) denotes the ith
i1 i2 ib
• We define differential error correcting index code in column of A. For a set of column indices I, AI denotes the
which the receivers have different error correcting submatrix of A with columns indexed by I. Similarly A(j)
capability. The necessary and sufficient conditions for denotes the jth row of A and for a set of row indices J, A
a matrix L to correspond to a differential index code denotes the submatrix of A with rows indexed by J. J
is found.
• We establish a link between linear differential error II. DIFFERENTIALERRORCORRECTINGINDEXCODES
correctingindexcodesandcertainmatroids.Wedefine ANDAUSEFULLEMMA
amatroidaldifferentialerrorcorrectingindexcodeand
In this section, we review error correcting index codes,
show that a linear differential error correcting index
define differential error correcting index code and establish a
code exists if and only if it is matroidal differential
lemmawhichweuseinSectionIVtoestablishourmainresult.
error correcting index code associated with a repre-
sentable matroid.
Error correcting index codes consider the scenario in
• We consider two special cases, δ-error correcting which the symbols received by receiver Ri may be subject
indexcodesanderrorcorrectionatasubsetofreceiver to errors. The source S broadcasts a vector C(x) ∈ FNq . The
nodes. The representable matroids associated with error affecting receiver Ri is considered as an additive error
these two special cases are identified. represented by (cid:15)i ∈FNq . Then, Ri actually receives the vector
y = C(x)+(cid:15) ∈ FN. An error correcting index code should
i i q
The organization of the paper is as follows. In Section be able to satisfy the demands of receivers in the presence of
II we review the definitions of error correcting index codes. these additive errors.
We define differential error correcting index codes and also
establishesalemmawhichisusedtoproveourmainresult.In Definition 2: Consider an instance of the index coding
Section III, basic results of Matroids are reviewed. Finally in problem described by (m,n,χ,f). A δ-error correcting index
Section IV, we define matroidal differential error correcting code(δ-ECIC)overFq forthisinstanceisanencodingfunction
index code and establishes the link between matroids and C : Fnq → FNq such that for receiver Ri, i ∈ (cid:100)m(cid:99), there
differential error correcting index codes. We conclude and exists a decoding function Di : FNq ×F|qχi| → Fq satisfying
summarize all the results in Section V. ∀ x∈Fn,∀ (cid:15) ∈FN,wt((cid:15) )≤δ :D (C(x)+(cid:15) ,x )=x .
q i q i i i χi f(i)
Notations: The size of a set S is denoted by |S|. Consider IftheencodingfunctionCislinearthenitisalinearerror
two sets S1 and S2. The set subtraction S1 \S2 is denoted correctingindexcode.Asforthelinearindexcode,linearerror
by S −S . The rank of a matrix A over F is denoted by
1 2 q correcting index code can also be described by a matrix L.
rank(A). For some positive integer N, identity matrix of size
N over F is denoted by I . The vector space spanned by Dau et al. in [5] identify a necessary and sufficient condi-
q N
columns of a matrix A over F is denoted by (cid:104)A(cid:105). Consider tion which a matrix L has to satisfy to correspond to a δ-error
q
correcting index code. Consider the set of vectors can be rewritten as
(cid:18) (cid:19)
I(q,H)(cid:44){z ∈Fnq :∃ i∈(cid:100)m(cid:99) | zχi =0 and zf(i) (cid:54)=0}. (z e) IsuLpp(e) (cid:54)=0,∀ z ∈Fnq :zχi =0,zf(i) (cid:54)=0, (4)
HereHisadirectedhypergraph[9]usedtodescribetheindex
∀ e∈F2δi,∀ supp(e)∈{F ⊆(cid:100)N(cid:99):|F|=2δ }.
coding problem (m,n,χ,f). The necessary and sufficient q i
conditionsforamatrixLtocorrespondtoaδ-errorcorrecting Let χ denote the set (cid:100)n(cid:99)−χ . Since at a particular receiver
i i
indexcodeisasfollows.ThematrixLcorrespondstoa(δ,H)- we consider only those z ∈ Fn for which z = 0 condition
ECIC over Fq if and only if (4) can be rewritten as q χi
wt(zL)≥2δ+1,∀ z ∈I(q,H). (1) (cid:18) L (cid:19)
(zχi e) I χi (cid:54)=0,∀ zχi ∈Fq|χi| :zf(i) (cid:54)=0,
supp(e)
The proof of (1) follows from the fact that if (1) is not ∀ e∈F2qδi,∀ supp(e)∈{F ⊆(cid:100)N(cid:99):|F|=2δi}.
satisfied then one can demonstrate the existence of a pair of (5)
informationvectorsxandx withx (cid:54)=x ,x =x and
(cid:48) f(i) (cid:48)f(i) χi (cid:48)χi We now present a lemma which will be used in Section IV to
a corresponding pair of error vectors (cid:15) and (cid:15)(cid:48) with wt((cid:15))≤δ prove the main result of this paper.
and wt((cid:15)) ≤ δ such that the corresponding received vectors
(cid:48)
atreceiverRi areequal.SothereceiverRi willnotbeableto Lemma 1: LetIf(i)denote(|χi|+2δi)lengthvectorwitha
distinguish between xf(i) and x(cid:48)f(i). Therefore, L corresponds f1(iin)oonfetohfethreecfieirvsetr|χRi|,poasnidtiownitchorarellspootnhderinegletomtehnetsde0m. aFnodr
to a δ-error correcting index code if and only if the following i
condition holds. For all i∈(cid:100)m(cid:99) and for all z ∈Fn such that some supp(e)∈{F ⊆(cid:100)N(cid:99):|F|=2δi} the condition
q
Tzχhiis=eq0uaatniodnzcfa(ni)b(cid:54)=er0ew, zriLtte+nien(cid:54)=ma0t,ri∀xefo∈rmFiNqn,thwet(feo)llo≤wi2nδg. (z e)(cid:18) IsuLpp(e) (cid:19)(cid:54)=0,∀ z ∈Fnq :zχi =0,zf(i) (cid:54)=0, (6)
way. For each receiver R
i ∀ e∈F2δi
(cid:18) (cid:19) q
L
(z e) IN (cid:54)=0,∀ z ∈Fnq :zχi =0,zf(i) (cid:54)=0, (2) holds if and only if the following condition holds
(cid:28)(cid:18) (cid:19)(cid:29)
∀ e∈FNq :wt(e)≤2δ. If(i) ⊆ I Lχi . (7)
supp(e)
ThematrixLcorrespondstoanerrorcorrectingindexcode Proof: Refer Appendix A.
if and only if (2) is satisfied at all receivers R , i∈(cid:100)m(cid:99).
i Lemma 1 gives an equivalent condition for equation (4).
Thereforeagivenindexcodeisdifferentialδ errorcorrecting
We now define a differential δ error correcting index R
R if and only if (7) holds for all supp(e) ∈ {F ⊆ (cid:100)N(cid:99) : |F| =
code. A differential error correcting index coding problem
2δ } and at all receivers R ,i∈(cid:100)m(cid:99).
i i
considers the scenario in which the error correcting capability
varies from receiver to receiver.
III. MATROIDS
Inthissectionwelistfewbasicdefinitionsandresultsfrom
Definition 3: Consider an instance of the index coding
matroid theory. These results and definitions are taken from
problemdescribedby(m,n,χ,f).Letδ ={δ ,δ ,...,δ },
R 1 2 N
[10].
where δ is the maximum number of errors receiver R wants
i i
to correct. A differential δR error correcting index code over Definition 4: Let E be a finite set. A matroid M on E
Fq for this instance is an encoding function C : Fnq → FNq is an ordered pair (E,I), where the set I is a collection of
such that for receiver Ri, i ∈ (cid:100)m(cid:99), there exists a decoding subsets of E satisfying the following three conditions
function Di : FNq ×F|qχi| → Fq satisfying ∀ x ∈ Fnq,∀ (cid:15)i ∈ (I1) φ∈I
FN,wt((cid:15) )≤δ :D (C(x)+(cid:15) ,x )=x .
q i i i i χi f(i)
(I2) If X ∈I and X ⊆X, then X ∈I.
(cid:48) (cid:48)
Equation (2) needs to be modified for the differential δ
R (I3) If X and X are in I and |X | < |X |, then there
error correcting index code. A matrix L corresponds to a 1 2 1 2
is an element e∈X −X such that X ∪e∈I.
differential error correcting index code if and only if the 2 1 1
following condition holds. For each receiver R ,i∈(cid:100)m(cid:99), The set E is called the ground set of the matroid and is
i
(cid:18) (cid:19) alsoreferredtoasE(M).ThemembersofsetI arecalledthe
(z e) L (cid:54)=0,∀ z ∈Fn :z =0,z (cid:54)=0, independent sets of M. Independent sets are also denoted by
IN q χi f(i) (3) I(M). A maximal independent subset of E is called a basis
∀ e∈FN :wt(e)≤2δ . ofMandthesetofallbasesofMisdenotedbyB(M).With
q i
M, a function called the rank function is associated, whose
domain is the power set of E and codomain is the set of non-
The error pattern corresponding to an error vector e is negative integers. The rank of any X ⊆E in M, denoted by
defined as its support set supp(e). Let I denote the r (X) is defined as the maximum cardinality of a subset X
supp(e)
submatrix of I consisting of those rows of I indexed by thMat is a member of I(M). The rank of matroid is the rank
N N
supp(e). For a receiver R , the error correcting condition (3) of its ground set.
i
Definition 5: Two matroids M and M are isomorphic, (A) g is one-one on (cid:100)n(cid:99), and g((cid:100)n(cid:99))∈I(M).
1 2
denotedasM ∼=M ,ifthereisabijectionψfromE(M )to
1 2 1 (B) For at least one basis B of M obtained by extending
E(M ) such that, for all X ⊆E(M ),ψ(X) is independent
2 1 g((cid:100)n(cid:99)), i.e. B −g((cid:100)n(cid:99)) = {b ,b ,...,b }, the
in M if and only if X is independent in M . n+1 n+2 n+N
2 1 following conditions should hold.
Definition 6: The vector matroid associated with a matrix
A over some field F, denoted by M[A], is defined as the (B1) g(ci)∈/ cl (B−g((cid:100)n(cid:99))),∀ i=1,2,...,N.
M
ordered pair (E,I) where E consists of the set of column
labelsofA,andI consistsofallthesubsetsofE whichindex
columns that are linearly independent over F. An arbitrary (B2) r (g((cid:100)n(cid:99))∪b ∪g(c ))=r (g((cid:100)n(cid:99))∪b )
n+i i n+i
matroid M is said to be F-representable if it is isomorphic M =rM(g((cid:100)n(cid:99))∪g(c )).
i
to a vector matroid associated with some matrix A over some M
field F. The matrix A is then said to be a representation of
(C) For each receiver R and for each error pattern F =
M. A matroid which is not representable over any finite field i
{e ,e ,...,e }, let
is called a non-representable matroid. i1 i2 i2δi
B =B−g(χ )−{b ,b ,...,b }.
ConsideramatroidMwithmatrixAasitsrepresentation. F,i i n+i1 n+i2 n+i2δi
Each element in ground set of M corresponds to a column
LetM bethe|χ |+N+2δ elementmatroidM/B .
in A. For a subset S of ground set E(M), AS denotes the ThenaFte,iveryreceivierR andiforeachvaliderrorpattFer,in
i
submatrix of A with columns corresponding to the elements F we must have
of ground set in S.
r (g({c ,c ,...,c })∪g(f(i)))
Lemma 2: Let M be a representable vector matroid with MF,i 1 2 N
A as its representation matrix over field F. The matroid M =rMF,i(g({c1,c2,...,cN})).
remains unchanged if any of the following operations are
performed on A Definition 8 can be viewed as a matroidal abstraction of
differential δ error correcting index codes. Condition (A)
R
• Interchange two rows. ensures that the vectors representing the messages are linearly
• Multiply a row by a non-zero member of F. independent. Condition (B) is equivalent to the condition
that scalar linear index code should be a non-zero linear
• Replace a row by the sum of that row and another. combination of messages added with a linear combination
of errors. Condition (C) ensures that the receivers can de-
• Adjoin or delete a zero row.
code their demands in the presence of errors. Condition (C)
• Multiply a column by a non-zero member of F. requires that g({c ,c ,...,c }) and g(f(i)) is present in
1 2 N
E(M ). However this is ensured because condition (B)
Definition 7: Let M = (E,I) be a matroid and T ⊆ E. ensureFs,ig({c ,c ,...,c })doesnotbelongtoB andhenceit
ConsiderthesetI ={I ⊆E−T :I∪B ∈I},whereB ⊆ 1 2 N
(cid:48) T T isnotcontractedout.Alsosinceg(f(i))⊆g(χ ),itisalsonot
T is a maximal independent subset within T. The contraction i
contracted out of the matroid M. We now present the main
of T from M, denoted as M/T, is the matroid (E−T,I ).
(cid:48) resultofthispaperwhichrelatesscalarlineardifferentialerror
The contraction of a F-representable matroid is also F- correcting index codes to representable matroids.
representable.LetM[A]bethevectormatroidassociatedwith
Theorem 1: Let (m,n,χ,f) be an index coding problem.
amatrixAoverF.Letebetheindexofanon-zerocolumnof
Ascalarlineardifferentialδ errorcorrectingindexcodeover
A.Supposeusingtheelementaryrowoperations,wetransform F existsifandonlyiftheiRndexcodeismatroidaldifferential
A to obtain a matrix A(cid:48) which has a single non-zero entry in δq errorcorrectingassociatedwithaF representablematroid.
column e. Let A denote the matrix which is obtained by R q
(cid:48)(cid:48)
deleting the row and column containing the only non-zero Proof: Refer Appendix B.
entry of column e. Then M[A]/{e}=M[A ].
(cid:48)(cid:48) Theorem1establishesalinkbetweenscalarlineardifferen-
tialδ errorcorrectingindexcodesandarepresentablematroid
IV. MATROIDALERRORCORRECTINGINDEXCODES satisfRying certain properties. In the examples below, we con-
In this section we first define matroidal differential error sider differential error correcting index coding problems with
correcting index codes and then prove the main result of the a scalar linear solution and show the representable matroid
paper. associated with it.
Definition 8: Consider an index coding problem Example 1: Let q = 2,m = n = 3 and f(i) = i,∀ i ∈
(m,n,χ,f) with an index code {c1,c2,...,cN} of length N. (cid:100)m(cid:99). Let χ1 ={2},χ2 ={1,3},χ3 ={2,1}. Let δ1 =2 and
Let δR ={δ1,δ2,...,δN}, where δi is the maximum number δ2 = δ3 = 1. Consider the index code C of length N = 7
of errors receiver R wants to correct. Let M = (E,I) be a described by the matrix described by the matrix
i
matroid defined over a ground set E with n+2N elements (cid:34) 1 1 1 1 0 1 0 (cid:35)
and with r(M) = n+N. The index code {c ,c ,...,c }
1 2 N L= 0 0 1 0 1 1 0 .
is said to be matroidal differential δ error correcting
R 0 1 0 1 1 0 1
index code associated with M, if there exists a function
g : (cid:100)n(cid:99)∪{c ,c ,...,c } → E(M) such that the following We have C(x)={x ,x +x ,x +x ,x +x ,x +x ,x +
1 2 N 1 1 3 1 2 1 3 2 3 1
conditions are satisfied. x ,x }. The index code C can also be described by seven
2 3
encoding functions S(C) = {c ,c ,...,c }. Consider the is a matroidal differential δ error correcting index code. It
1 2 7 R
representable matroid M associated with 10 × 17 matrix can be verified that the index code C is differential δ error
R
(cid:34) L (cid:35) correcting.
A= I . The function g :(cid:100)3(cid:99)∪{c ,c ,...,c }→
10 1 2 N
I Example 2: Let q = 2,m = n = 5 and f(i) = i,∀ i ∈
7
E(M), is defined as follows, (cid:100)m(cid:99). Let χ1 = {2,5},χ2 = {1,3},χ3 = {2,4},χ4 = {3,5}
and χ = {1,4}. Let δ = δ = δ = 1 and δ = δ = 2.
5 1 4 5 2 3
g(i)=i,∀ i∈(cid:100)3(cid:99) Consider the index code C of length N =8, described by the
matrix
g(ci)=10+i,∀ i∈(cid:100)7(cid:99).  1 1 1 1 0 0 0 0 
The matroid M and function g satisfies Definition 8. The  0 1 1 1 0 1 1 0 
first three columns corresponding to messages are linearly L= 1 1 0 0 1 1 1 0 .
independent. Thus Condition (A) is satisfied. Consider the  0 1 0 0 1 0 1 1 
basis B = {b ,b ,...,b } = (cid:100)10(cid:99). The set B −g((cid:100)3(cid:99)) = 1 0 0 1 0 0 1 1
1 2 10
{4,5,...,10}.Thevectorscorrespondingtog(c )doesnotlie
i The index code C can be viewed as eight encoding functions
inthecolumnspaceofthematrixA(B−g((cid:100)3(cid:99))).ThusCondition S(C)={c ,c ,...,c }. Consider a representable matroid M
1 2 8
(B1) is satisfied. Condition (B2) is actually seven conditions
associated with 13×21 matrix
corresponding to each c ,i∈(cid:100)7(cid:99). For c , we have,
i 1 (cid:34) L (cid:35)
 1 0 0 0 1  A= I .
13
 0 1 0 0 0  I8
A(g((cid:100)3(cid:99))∪b4∪g(c1)) = 0 0 1 0 0 ,
 0 0 0 1 1  The function g :(cid:100)5(cid:99)∪S(C)→E(M), is defined as follows,
O
g(i)=i,∀ i∈(cid:100)5(cid:99)
where O is a 6×6 all zero matrix. It is clear that the fifth
g(c )=13+i,∀ i∈(cid:100)8(cid:99).
columnofAg((cid:100)3(cid:99))∪b4∪g(c1)) isalinearcombinationofthefirst i
four columns and also the fourth column is a linear combina- One can verify that the matroid M and function g satisfies
tion of the remaining four columns. Similarly condition (B2) Definition8.Notethatthelengthofthisindexcodeiseight.In
can be verified for all c ,i∈(cid:100)7(cid:99). [5], for this index coding problem with all receivers requiring
i
double error correction, the optimal length of index code was
Condition (C) needs to be verified at all receivers. Let found to be 9. However since here few receivers require only
us consider receiver R1 with δ1 = 2. Consider an error single error correcting capability we are able to find the code
gupMsa(eχt/tde1Br)nfF−o1r,F1{r,141e,,p5r=ie,ss6e,an7{t}eav1tei=,ocent2o{,ro2ef,3m8,t,eah94ter},o1.mi0dWa}.t.reLoTeihdhteaMMvmeia,jtBroFbi1de.,1MTthhe=eF1m,v1Be,a1cttro=−ixr onrmefepaeltrdereonstigeodnthfitiasnetdiriogetndhhsute.corTefedhpsertmeorsaee2lsnl1tetr.raictFsitooiizonrena.asrTdeohonleuetbshssliee.zTemehroarifostrrotehnicdeoabrgfrloreeorcsutwiunnshdgitcosihnefitdwneodxef
representation for this matroid is given by thFei,jm,iatrix code the size of the ground set is 23.
 1 1 1 1 0 1 0 
0 1 0 1 1 0 1
 1 0 0 0 0 0 0  A. δ-Error Correcting Index Codes
 
M1,1 = I6 . Here we consider δ-error correcting index codes in which
 0 1 0 0 0 0 0 
 0 0 1 0 0 0 0  all the receivers have the ability to correct δ number of errors.
This is a special case of differential δ error correcting index
0 0 0 1 0 0 0 R
code in which δ = δ,∀i ∈ (cid:100)m(cid:99). We consider a single
i
InordertosatisfyCondition(C),wewantthecolumnsofM error correcting index coding problem and show the matroids
1,1
correspondingtog(f(1))tobeinthelinearspanofcolumnsof associated with it in Example 3 in Appendix C.
M corresponding to g(S(C)). The corresponding columns
1,1
are given below:
 1   1 1 1 1 0 1 0  B. Error Correction at only a subset of Receivers
0 0 1 0 1 1 0 1
   
Mg(f(1)) = 0 ,Mg(S(C)) = 1 0 0 0 0 0 0 . codeNsowwitwheercroonrscidoerrrecatniontghecrapsapbeicliiatyl caatsaesoufbseerrtoorfcroercreeicvteinrsg.
1,1  0  1,1  0 1 0 0 0 0 0 
    Consider a subset S of (cid:100)m(cid:99). Each receiver R ,i ∈ S should
 0   0 0 1 0 0 0 0  i
be able to correct δ errors. We can obtain error correcting
0 0 0 0 1 0 0 0 i
only at particular subset of receivers, from a differential error
We can observe that columns corresponding to g(f(1)) lies correctingindexcodebysettingδ =0,∀i∈/ S.Notethataδ-
i
in the linear span of columns corresponding to g(S(C)). errorcorrectingindexcodecorrectsδ errorsatallreceivers.So
Condition (C) is verified for receiver R and error pattern it obviously corrects errors at a subset of receivers. However
1
{e ,e ,e ,e }.Similarlyitcanbeverifiedforallerrorpatterns since the conditions which the matroid to satisfy are less we
1 2 3 4
and for all receivers. Thus the matroid M with representation will be able to find other representations for the matroid. We
A and function g satisfies Definition 8. So the index code C illustrate this in Example 4 in Appendix D.
(cid:18) (cid:19)
V. CONCLUSION of I Lχi should generate If(i). There should be some
supp(e)
In this paper we have defined differential error correcting N ×1 vector X such that,
indexcodeswhichisageneralizationoferrorcorrectingindex
(cid:18) (cid:19)
codes. We established the connections between differential I Lχi X =If(i).
error correcting index codes and matroids. It was shown that supp(e)
scalar linear differential error correcting codes correspond to
Now suppose for some (z e), with z (cid:54)= 0,z = 0 and
representable matroid with certain properties. This sheds lots f(i) χi
some e∈F2δi we have
of light in construction and existence of differential error q
correcting codes for an index coding problem. The definition (cid:18) (cid:19)
L
of a matroidal differential error correcting index code is more (z e) I =0.
supp(e)
general and the matroid need not be a representable matroid.
Using a non representable matroid satisfying the conditions, Since z =0, the above equation reduces to
χi
the possibility of non linear differential error correcting codes (cid:18) (cid:19)
L
could be explored. (zχi e) I χi =0.
supp(e)
REFERENCES Multiplying both sides by X, we get zf(i) = 0, which is a
contradiction. This completes the if part.
[1] Y. Birk and T. Kol,“Informed-source coding-on-demand (ISCOD)
Now we prove the only if part. Let L denote the row of L
overbroadcastchannels,”inProc.IEEEConf.Comput.Commun.,San f(i)
Francisco,CA,1998,pp.1257-1264. corresponding to the message demanded by receiver Ri. Let
[2] Z.Bar-Yossef,Z.Birk,T.S.JayramandT.Kol,“Indexcodingwith ri denotetheset(cid:100)n(cid:99)−χi−f(i).LetLri denotethesubmatrix
side information,” in Proc. 47th Annu. IEEE Symp. Found. Comput. of L with rows indexed by the set r . Because (5) holds, we
i
Sci.,2006,pp.197-206. have
[3] S. E. Raouayheb, A. Sprinston and C. Georghiades, “On the Index  
codingproblemanditsrelationtonetworkcodingandmatroidtheory,” (cid:18) L (cid:19) Lf(i)
[4] iVn.ITE.EMEuTrarlaindsh.arIannf.aTnhdeoBr.y,Sv.oRla5ja6n,,n“oL.i7n,eparpN.e3t1w8o7r-k31C9o5d,iJnugl,yL2i0n1ea0r. rank Isupχpi(e) = rank IsuLprpi(e) 
index coding and representable discrete polymatroids,”, Available on (cid:18) (cid:19)
ArXivathttp://arxiv.org/abs/1306.1157. = rank(cid:0)Lf(i)(cid:1)+rank I Lri
[5] SonHoangDau,VitalySkachekandYeowMengChee,“ErrorCorrec- supp(e)
tion for Index Coding with Side Information,” in IEEE Transactions (cid:18) L (cid:19)
onInformationTheory,Vol.59,No.3,March2013. = 1+rank I ri .
[6] R. W. Yeung and N. Cai, “Network error correction, part I: basic supp(e)
concepts and upper bounds,” in Comm. in Inform. and Systems, vol.  
L
6,2006,pp.19-36. χi
[7] R. W. Yeung and N. Cai, “Network error correction, part II: lower Consider the concatenated matrix  If(i) , de-
I
bounds,”inComm.inInform.andSystems,vol.6,2006,pp.37-54. supp(e)
[8] K.PrasadandB.S.Rajan,“AMatroidalframeworkforNetwork-Error noted by Y. We have,
CorrectingCodes,”IEEETransactionsonInformationTheory,Vol.61, (cid:18) (cid:19)
No.2,pp.836-872,February2015. rank(Y) = rank(cid:0) Lf(i) 1 (cid:1)+rank I Lri
[9] N.Alon,A.hassidimi,E.Lubetzky,U.StavandA.Weinsten,“Broad- supp(e)
castingwithsideinformation,”inProc.49thAnnu.IEEESymp.Found. (cid:18) L (cid:19)
Comput.Sci.,2008,pp.823-832. = 1+rank I ri .
[10] J.G.Oxley,“MatroidTheory”,OxfordUniversityPress,1992. supp(e)
The concatenated matrix Y has the same rank as the matrix
(cid:18) (cid:19)
L
APPENDIXA I χi . This proves the only if part.
PROOFOFLEMMA1 supp(e)
Lemma1:LetI denote(|χ |+2δ )lengthvectorwitha
f(i) i i
1 in one of the first |χ | position corresponding to the demand
i APPENDIXB
f(i) of the receiver R , and with all other elements 0. For
i PROOFOFTHEOREM1
some supp(e)∈{F ⊆(cid:100)N(cid:99):|F|=2δ } the condition
i
(cid:18) (cid:19) Theorem 1: Let (m,n,χ,f) be an index coding problem.
L
(z e) Isupp(e) (cid:54)=0,∀ z ∈Fnq :zχi =0,zf(i) (cid:54)=0, (8) AF secxailsatrsliinfeaanrddoifnfleyreinfttihaleδiRndeerxrocrocdoerirsecmtiantgroiinddaelxdcifofedreenotviearl
q
∀ e∈F2qδi δR errorcorrectingassociatedwithaFq representablematroid.
holds if and only if the following condition holds Proof:Firstweprovetheonlyifpart.Supposethereexists
a scalar linear differential error correcting index code C of
(cid:28)(cid:18) (cid:19)(cid:29)
If(i) ⊆ I Lχi . (9) length N over Fq for the index coding problem (m,n,χ,f).
supp(e) Since the index code is linear it is represented by a n×N
(cid:18) (cid:19)
L
matrix L. Let ζ be the concatenated matrix . Note
Proof: The if part is proved first. Since I is in I
(cid:28)(cid:18) (cid:19)(cid:29) f(i) N
L that order of ζ is (n+N)×N. Let Y be the column wise
subspaceof I χi ,linearcombinationsofcolumns concatenatedmatrix( I ζ )ofsize(n+N)×(n+2N).
supp(e) n+N
Let M = M(Y), the vector matroid associated with Y, with also ensures that g(c ) ∈/ B. The vector representing g(c ) is
j j
E(M) being the set of column indices of Y. Clearly rank of Y(g(cj)). So we have
matroid M is n+N. The function g :(cid:100)n(cid:99)∪S(C)→E(M) (cid:88)
is defined as follows, Y(g(cj)) = ai,jY(g(i))+djY(n+j),
i n
g(i)=i,∀ i∈(cid:100)n(cid:99) ∈(cid:100) (cid:99)
for some a and d in F . As Condition (B1) holds, at least
i,j j q
g(ci)=n+N +i,∀ i∈(cid:100)N(cid:99). oneoftheai,j (cid:54)=0,∀i∈(cid:100)n(cid:99).Condition(B2)alsoensuresthat
d (cid:54)= 0,∀ j ∈ (cid:100)N(cid:99). This also ensures that g(c ) (cid:54)= g(c ) for
We have to show that the matroid M and function g satisfies j i j
the conditions of Definition 8. Clearly g(i) is one-one for distinct ci,cj ∈ C. Arranging all Y(g(cj)), we get Yg(C) = ζ,
and
i ∈ (cid:100)n(cid:99) and g((cid:100)n(cid:99)) corresponds to the first n columns of (cid:18) (cid:19)
L
Y which are columns of identity matrix. So g((cid:100)n(cid:99)) ∈ I(M) ζ = Kn×N ,
which proves Condition (A). N×N
Consider the basis of M, B = {1,2,...,n+N} obtained whereLcomprisesoftheelementsa ,1≤i≤n,1≤j ≤N
i,j
by extending g((cid:100)n(cid:99)). From the definition of g, we have and K is a diagonal matrix with d ,1 ≤ j ≤ N as its
j
g(c ) = n+N +i and the corresponding vector associated diagonal entries. The matroid M does not change if some
i
with it is Y(n+N+i). From the definition of Y, we have row or some column of its representation is multiplied by a
(cid:18) L(i) (cid:19) non-zero element of F . The matrix Y is now of the form
Y(n+N+i) = ζ(i). Note that ζ(i) = , and ζ(i) does q
I(i) Y = ( In+N ζ ). Consider the matrix Y(cid:48) obtained from Y
notbelongtothelinearspanofYB g( nN) becauseL(i) isnon by multiplying the rows {n + 1,n + 2,...,n + N} by the
zero. Column L(i) corresponds to −the(cid:100)it(cid:99)h transmission of the elements {d−11,d−21,...,d−N1} respectively and then multiply-
indexcodeandforascalarlinearcodeitisanon-triviallinear ing columns {n+1,n+2,...,n+N} by {d1,d2,...,dN}
respectively.ThematrixY isoftheform( I ζ )where
combination of source messages. So condition (B1) holds. (cid:48) n+N (cid:48)
The vector ζ(i) lies in the linear span of (cid:16) In(cid:100)n+(cid:99)N I(nn++iN) (cid:17) ζ(cid:48) = (cid:18) LIn×N (cid:19). The matrix Y(cid:48) is a representation for the
N
because L(i) lies in the linear span of I(cid:100)n(cid:99) and the vector matroid M proving our claim. In the last part of the proof
n+N
I(n+i) has a 1 at (n+i)th position. So condition (B2) holds. we show that the matrix L corresponds to a differential error
n+N correcting index code.
Consider a receiver R and an error pattern F =
i j
{e ,e ,...,e }. Let I(F ) = {i ,i ,...i } be the set Consider a receiver Ri and an arbitrary error pattern Fj. For
i1 i2 i2δi j 1 2 2δi this the matroid M is a vector matroid of the matrix
of indices corresponding to the error pattern and let the set Fj,i
{n+i ,n+i ,...n+i } be denoted as n+I(F ). From (cid:0) (cid:1)
tthhee mdeafi1trniixtion o2f MFj,i w2δei note that it is a vector mjatroid of wZh=ereYgI(cid:48)((χFi)∪)(ni+sIt(hFej)i)n=dex sIegt(χcio)∪rr(ens+pIo(nFdj)in)g ζtog(cid:48)(χeir)r∪o(rn+paIt(tFejr)n) ,
Ntoogte(Sth(aCt)f)roism the dZefi=nitYiogn(χoi)f∪g(n,+thIe(Fmj)a)t.rix that corresponds F(Cj),awnde hζag(cid:48)vj(χei)Z∪((ng(+fI((i)F)j)))⊆=(cid:104)Zg(cid:18)(S(LICIg)()(F(cid:105)χ.ji))N(cid:19)ot.eFthroatmZCg(oSn(Cd)it)io=n
ζ and that Z(g(f(i))) = I . Thus I ⊆
SinceweZhagv(Se(aC)s)c=alaζrgl(iχnie)a∪r(nd+ifIf(eFre))nt=ial(cid:18)errIoILr(χFcijo)rre(cid:19)ct.ingindex (cid:18)wg(cid:48)a(sχLIiIga)(∪(rFχb(jini))t+raI(cid:19)r(yF.,jA)u)ssinthgeLcehmoimceao1firteciseivseeernantfhd(ai)tthteheerirnodrexpf(aict)toedrne
code, from Lemma 1, we have given by the matrix L is differential error correcting. This
completes the proof of the theorem.
(cid:28)(cid:18) (cid:19)(cid:29)
L
I ⊆ χi .
f(i) I
supp(e) APPENDIXC
Notethatf(i)∈(cid:100)n(cid:99)andthereforeg(f(i))=f(i).Sowehave EXAMPLEOFANERRORCORRECTINGINDEXCODE
Z(g(f(i))) = Z(f(i)) which is equal to I . Thus Z(g(f(i)))
f(i) Example 3: Let q = 2,m = n = 3 and f(i) = i for
liesinthelinearspanofZg(S(C)) andCondition(C)holdsfor
i ∈ (cid:100)3(cid:99). Let χ = {2,3},χ = {1,3} and χ = {1,2}.
error pattern F and receiver R . Since the receiver and error 1 2 3
j i Considerthescalarlinearindexcoderepresentedbythematrix
pattern was chosen arbitrarily this completes the only if part
of the proof. (cid:34) 1 1 1 (cid:35)
L= 1 1 1 .
Now we have to prove the if part. Let M be the given F
q 1 1 1
representablematroidwithfunctiong andbasisB =g((cid:100)n(cid:99))∪
{b ,b ,...b }, that satisfy the set of conditions. Let Let C be the index code based on L. We have C(x)={x +
n+1 n+2 n+N 1
Y = ( I ζ ) be a representation of M over F such x +x ,x +x +x ,x +x +x } and S(C)={c ,c ,c }.
n+N q 2 3 1 2 3 1 2 3 1 2 3
that B = {1,2,...,n+N}. First we prove that there exists It was shown in [5] that the index code corresponding to L is
(cid:18) (cid:19)
L capableofcorrectingsingleerror.Weshowthattheindexcode
an n×N matrix L such that ζ = .
I isamatroidaldifferentialδ errorcorrectingindexcodeasso-
N R
Consider a symbol in the index code c ∈ C. From condition ciated with a F representable matroid, where δ ={1,1,1}.
j 2 R
(B2)wehavethatthecolumnvectorrepresentingg(c )liesin Inordertoshowthatwefirstspecifytherepresentablematroid
j
the linear span of vectors representing g((cid:100)n(cid:99)) and b . This and show the function mapping messages and transmitted
n+i
symbols to the ground set of matroid. Consider the vector (C) is verified for receiver R and error pattern {e ,e }. Let
1 1 2
matroid M associated with the 6×9 matrix usconsideranothererrorpatternF ={e ,e }.Thematroid
1,2 1 3
  M = M/B is the matroid which we have to
 10 01 00 00 00 00 11 11 11  conFs1id,2e,1r for error paFt1t,e2r,n1 F1,2 and at Receiver R1. The vector
A= 0 0 1 0 0 0 1 1 1 . representation of the matroid MF1,2,1 is given by the matrix
 0 0 0 1 0 0 1 0 0  (cid:34) 1 0 0 1 1 1 (cid:35)
 0 0 0 0 1 0 0 1 0  M = 0 1 0 1 0 0 .
0 0 0 0 0 1 0 0 1 1,2
0 0 1 0 0 1
The edge set of matroid, E(M) is (cid:100)9(cid:99). The function g :(cid:100)3(cid:99)∪ We can verify that the first column of matrix M can be
1,2
S(C)→E(M), is defined as follows, obtained as the linear combination of last three columns of
matrix. The last error patter to consider is F = {e ,e }.
g(i)=i,∀ i∈(cid:100)3(cid:99) 1,3 2 3
Here we consider the matroid M , whose representation
g(c )=6+i,∀ i∈(cid:100)3(cid:99). is given by the matrix F1,3,1
i
We show that matroid M and function g satisfies conditions (cid:34) 1 0 0 1 1 1 (cid:35)
of Definition 8. The first three columns corresponding to the M = 0 1 0 0 1 0 .
1,3
messages are linearly independent. This satisfies Condition A. 0 0 1 0 0 1
Let us consider the basis B = (cid:100)6(cid:99). The set B −g((cid:100)3(cid:99)) =
The column corresponding to g(f(1)) is the first column and
{4,5,6}. The vector corresponding to g(c ) does not lie in
i the columns corresponding to g(S(C)) are last three columns.
the column space of the matrix AB g( 3 ). Thus Condition
− (cid:100) (cid:99) The linear dependence is very clear in this scenario also.
(B1) is satisfied. Condition (B2) is actually three conditions
Condition(C)canbeverifiedforotherreceivers.Thematrices
corresponding to each c ,i∈(cid:100)3(cid:99). For c , we have,
i 1 corresponding to various error patterns at other receivers are
 1 0 0 0 1  given in Table I. Condition (C) at all receivers can be verified
0 1 0 0 1 using Table I.
 
 0 0 1 0 1 
A(g((cid:100)3(cid:99))∪b4∪g(c1)) = 0 0 0 1 1 . APPENDIXD
 0 0 0 0 0  EXAMPLEOFANINDEXCODECAPABLEOFERROR
0 0 0 0 0
CORRECTINGATASUBSETOFRECEIVERS
It is clear that the fifth column of A(g((cid:100)3(cid:99))∪b4∪g(c1)) is a Example 4: In this example we consider the index coding
linear combination of the first four columns and also the problemofExample3.Howeverinthisexampleonlyreceiver
fourth column is a linear combination of the remaining four R requires single error correction. Consider another index
1
columns. Similarly condition (B2) can be verified for c and code C described by the matrix
2 2
c .Condition(C)needstobeverifiedforvariouserrorpatterns.
3 (cid:34) 1 1 1 (cid:35)
Let F represent the jth error patter at receiver R . For
i,j i L = 1 1 0 .
example the error pattern F = {e ,e } is the first error 1
1,1 1 2 1 0 1
pattern at receiver R . Note that δ = 1 for all receivers as
1 i
we are using a single error correcting code. For receiver R1, We have C2(x)={x1+x2+x3,x1+x2,x1+x3}. Consider
χ ={1}. We have, therepresentablematroidM associatedwiththe6×9matrix
1 (cid:48)
 
B =B−g(χ )−{4,5} 1 0 0 0 0 0 1 1 1
F1,1,1 1 0 1 0 0 0 0 1 1 0
={2,3,6}.  
 0 0 1 0 0 0 1 0 1 
A = .
The matroid M = M/B , is a vector matroid. Let 1  0 0 0 1 0 0 1 0 0 
M be the maFtr1i,1x,1used for Fre1,p1r,1esentation of the matroid  0 0 0 0 1 0 0 1 0 
i,j
M . The vector representation for this matroid is given 0 0 0 0 0 1 0 0 1
byFthi,ej,imatrix
The function g :(cid:100)3(cid:99)∪S(C )→E(M), is defined as follows,
2
(cid:34) 1 0 0 1 1 1 (cid:35)
g(i)=i,∀ i∈(cid:100)3(cid:99)
M = 0 1 0 1 0 0 .
1,1
0 0 1 0 1 0 g(c )=6+i,∀ i∈(cid:100)3(cid:99).
i
InordertosatisfyCondition(C),wewantthecolumnsofM1,1 Condition A and condition B of Definition 8 holds true for
correspondingtog(f(1))tobeinthelinearspanofcolumnsof this matroid. Let us consider receiver R , basis B =(cid:100)6(cid:99) and
1
M1,1 corresponding to g(S(C)). The corresponding columns error pattern F ={e ,e }. For receiver R , χ ={1}. We
1,1 1 2 1 1
are given below:
have,
(cid:34) 1 (cid:35) (cid:34) 1 1 1 (cid:35)
B =B−g(χ )−{4,5}
M1g,(1f(1)) = 00 ,M1g,(1S(C)) = 10 01 00 . F1,1,1 ={2,3,6}.1
lIitniesarclsepaarnthoaftccoolluummnnssccoorrrreessppoonnddiinnggttoogg((Sf((C1)))).lCieosnidnititohne MThie,jmbaetrtohiedreMpr(cid:48)Fes1e,1n,1ta=tionMo(cid:48)f/tBhFe1m,1,a1triosidaMvec(cid:48)Ftio,jr,im. Tathreoivde.cLtoert
(i,j) Mi,j Mif,(ji) Mig,(jC(x)) (i,j) Mi,j Mif,(ji) Mig,(jC(x))
           
1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1
(1,1)  0 1 0 1 0 0   0   1 0 0  (1,1)  0 1 0 1 0 0   0   1 0 0 
0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0
           
1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1
(1,2)  0 1 0 1 0 0   0   1 0 0  (1,2)  0 1 0 1 0 0   0   1 0 0 
0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1
           
1 0 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1
(1,3)  0 1 0 0 1 0   0   0 1 0  (1,3)  0 1 0 0 1 0   0   0 1 0 
0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1
      TABLEII. MATRICESREPRESENTINGTHECONTRACTEDMATROIDSIN
(2,1)  100 010 001 110 101 100   100   110 101 100  EXAMPLE4FORRECEIOVFETRHRE1M.AMTARTORIDIXMM(cid:48)Fi,ij,jI,SiTHEREPRESENTATION
     
1 0 0 1 1 1 1 1 1 1
(2,2)  0 1 0 1 0 0   0   1 0 0  In order to satisfy condition (C) we want the columns of
0 0 1 0 0 1 0 0 0 1
M corresponding to g(f(2)) to be in the linear span of
2,1
columns of M corresponding to g(S(C )). Observe that
      2,1 2
1 0 0 1 1 1 1 1 1 1 (cid:34) 1 (cid:35) (cid:34) 1 1 0 (cid:35)
(2,3)  00 10 01 00 10 01   00   00 10 01  Mg(f(2)) = 0 and Mg(S(C2)) = 1 0 0 . The
2,1 2,1
0 0 1 0
 1 0 0 1 1 1   1   1 1 1  rank of the columns corresponding to g(S(C2)) is 2 where
(3,1)  0 1 0 1 0 0   0   1 0 0  as the rank of the columns corresponding to g(S(C2)) along
0 0 1 0 1 0 0 0 1 0 with g(f(2)) is 3. Thus condition (C) is violated. So the
index code corresponding to matrix L does not give error
1
      correcting property at receiver R . Similarly for receiver R
1 0 0 1 1 1 1 1 1 1 2 3
(3,2)  0 1 0 1 0 0   0   1 0 0  and error pattern F3,2 ={e1,e3}, condition (C) gets violated.
0 0 1 0 0 1 0 0 0 1 The matrix M representing the matroid M is
3,2 (cid:48)F3,2,3
      (cid:34) 1 0 0 1 0 1 (cid:35)
1 0 0 1 1 1 1 1 1 1
(3,3)  0 1 0 0 1 0   0   0 1 0  M3,2 = 0 1 0 1 0 0 .
0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1
TABLEI. MATRICESREPRESENTINGTHECONTRACTEDMATROIDS We can observe that condition (C) is not satisfied by receiver
USEDINEXAMPLE1.MATRIXMi,j ISTHEREPRESENTATIONOFTHE R . Thus the index code described by L has single error
MATROIDMFi,j,i co3rrecting property only at receiver R . 1
1
representation of the matroid M is given by the matrix
(cid:48)F1,1,1
(cid:34) 1 0 0 1 1 1 (cid:35)
M = 0 1 0 1 0 0 .
1,1
0 0 1 0 1 0
WecanclearlyseethatCondition(C)issatisfied.Thematrices
representing the contracted matroids for receiver R is given
1
in Table II. Let us consider Receiver R , basis B = (cid:100)6(cid:99) and
2
error pattern F ={e ,e }. For receiver R , χ ={2}. We
2,1 1 2 2 2
have,
B =B\g(χ )\{4,5}
F2,1,2 2
={1,3,6}.
The matroid M =M/B is a vector matroid. The
vector representa(cid:48)Ft2i,o1n,2of the(cid:48)maFtr2o,1id,2M is given by the
matrix (cid:48)F2,1,1
(cid:34) 1 0 0 1 1 0 (cid:35)
M = 0 1 0 1 0 0 .
2,1
0 0 1 0 1 0