Error Correction for Index Coding With Side Information - Spms

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1528 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 3, MARCH 2013Definition 8.1: Letbe a set of instancesforanICSIproblem.A -error-correcting index code overis said to be static under the set if it is a -error-correcting-IC over for all instances .Recall that an instancecan be described by theside information hypergraph .Foraset of instances,letwhereis defined as in (3). We also define(26)Lemma 8.2: The matrix corresponds to a -errorcorrectingindex code which is static under if and only ifConsider an instance.Forall,wehave and , and thuswe deduce thatHence by (3), the cardinality of each set inis at mostTherefore, due to (26), every set inhas at mostelements.It remains to show that every nonempty subset of whosecardinality is at most belongs to . Consider an arbitrary-subset of ,with .Consider an instancewithand .Sincewe havefor all and for all choices of .Proof: The proof follows from Definition 8.1 and Lemma3.8.Notice that when is used for an instancewith ,thenthelast rows of are simplydiscarded.One particular family of interest is , the family thatcontains all instances where each receiver owns at leastmessages as its side information. More formallyA -error-correcting index code which is static underwill provide successful communication between the sender andthe receivers under the presence of at most errors, despite apossible change of the collection of the side information sets ,a change of the set of receivers, and a change of the demandfunction, as long as each receiver still possesses at leastmessages.In the rest of this section, we assume that ,and .Definition 8.3: An matrix is said to satisfy the-Property if any nontrivial linear combination of at mostrows of has weight at least .Proposition 8.4: The matrix corresponds to a-error-correcting linear index code, which is static under, if and only if satisfies the -Property.Proof: Let be an matrix that satisfies the-Property. We show that this is equivalent to the conditionthat corresponds to a -error-correcting linear index code,which is static under . By Lemma 8.2, it suffices toshow that is the collection of all nonempty subsetsof , whose cardinalities are not greater than .The proof follows.B. Application: Weakly Resilient FunctionsIn this section, we introduce the notion of weakly resilientfunctions. Hereafter, we restrict the discussion to the binaryalphabet.The concept of binary resilient functions was first introducedby Chor et al. [24] and independently by Bennet et al. [25].Definition 8.5: A functionis called -resilientif satisfies the following property: when arbitrary inputs ofare fixed and the remaining inputs run through all the-tuples exactly once, the value of runs through everypossible output -tuple an equal number of times. Moreover, ifis a linear transformation, then it is called a linear -resilientfunction. We refer to the parameter as the resiliency of .The applications of resilient functions can be found in faulttolerantdistributed computing, quantum cryptographic key distribution[24], privacy amplification [25], and random sequencegeneration for stream ciphers [26]. Connections between linearerror-correcting codes and resilient functions were establishedin [24].Theorem 8.6 [24]: Let be an binary matrix. Then,is a generator matrix of a linear error-correcting code withminimum distance ifandonlyif is-resilient.Remark 8.7: Vectorial Boolean functions with certain propertiesare useful for design of stream ciphers. These propertiesinclude high resiliency and high nonlinearity (see, for instance,[26]). However, linear resilient functions are still particularly interesting,since they can be transformed into highly nonlinear resilientfunctions with the same parameters. This can be achieved
DAU et al.: ERROR CORRECTION FOR INDEX CODING WITH SIDE INFORMATION 1529by a composition of the linear function with a highly nonlinearpermutation (see [27] and [28] for more details).Below we introduce a definition of a -weakly -resilientfunction, which is a weaker version of a -resilient function.Definition 8.8: A function is called -weakly -resilient if satisfies the property that every set of coordinatesin the image of runs through every possible output -tuple anequal number of times, when arbitrary inputs of are fixedand the remaining inputs run through all the -tuplesexactly once.Remark 8.9: A -weakly -resilient functioncan be viewed as a collection of different -resilient functions, each such function is obtained by taking somecoordinates in the image of . Similarly to [24], consider ascenario, in which two parties are sharing a secret key, whichconsists of randomly selected bits. Suppose that at some momentout of the bits of the key are leaked to an adversary.By applying a -resilient function to the current -bit key, twoparties are able to obtain a completely new and secret key ofbits, without requiring any communication or randomness generation.However, if the parties use various parts of the key forvarious purposes, they may only require one of the -bit secretkeys (instead of the larger -bit key). In that case, a -weakly-resilient function can be used. By applying a -weakly -resilientfunction to the current -bit key, the parties obtain a setof different -bit keys, each key is new and secret (however,these keys might not be independent of each other).Theorem 8.10: Let be an binary matrix. Then,satisfies the -Property if and only if the functiondefined by is -weakly -resilient.Proof:1) Suppose that satisfies the -Property. Take any-subset .ByDefinition 8.3, the submatrixof is a generating matrix of the error-correcting codewith the minimum distance . By Theorem 8.6,the functiondefined byis -resilient. Since is an arbitrary -subset of ,thefunction is -weakly -resilient.2) Conversely, assume that the function is -weakly -resilient.Take any subset , . Then, the functiondefined by is -resilient.Therefore, by Theorem 8.6, is a generating matrixof a linear code with minimum distance .Sinceis an arbitrary -subset of ,byProposition8.4, satisfiesthe -Property.C. Bounds and ConstructionsIn this section, we study the problem of constructing a matrixsatisfying the -Property. Such with the minimal possiblenumber of columns is called optimal. First, observe thatfrom Proposition 8.4, we havewhich is the set of all nonempty subsets of of cardinality atmost . Next, consider an instance satisfying(27)whereis the side information hypergraphcorresponding to that instance. Such an instance can beconstructed as follows. For each subset, we introduce a receiver which requests the messageand has a set as its side information.It is straightforward to verify that indeed we obtain an instancesatisfying (27). The problem of designing anoptimal matrix satisfying the -Property then becomesequivalent to the problem of finding an optimal -ECIC.Thus, is equal to the number of columns in an optimalmatrix which satisfies the -Property.The corresponding -bound and -bound for canbe stated as follows.Theorem 8.11: Letlinearbe the smallest number such that acode exists. Then, we haveProof: The first inequality follows from the -bound andfrom the fact that , which is due to (27).For the second inequality, it suffices to show that. By Corollary 3.10, an matrix corresponds to an-IC if and only ifis linearly independent forevery .Since is the set of all nonemptysubsets of cardinality at most , this is equivalent to saying thatevery set of at most rows of is linearly independent. Thiscondition is equivalent to the condition that is a parity checkmatrix of a linear code with the minimum distance at least[29, Ch. 1]. Therefore, a linear -IC of length exists if andonly if an linear code exists. Since is thesmallest number such that ancode exists,we conclude that .Corollary 8.12: The length of an optimal -error-correctinglinear index code over which is static under satisfieswhere is the smallest number such that ancode exists.Proof: This is a straightforward corollary of Theorem 5.1(the Singleton bound) and Theorem 8.11.Corollary 8.13: For, the length ofan optimal -error-correcting linear index code over whichis static under is .Proof: For , there exists anlinear code with (for example, one can take an extendedRS code [29, Ch. 11]). Due to the Singleton bound, we concludethat is the smallest value such thatlinear code exists. Following the lines of the proof of Theorem8.11, there exists a -error-correcting index code of length, which is static under .As ,we have
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