Error Correction for Index Coding With Side Information - Spms

DAU et al.: ERROR CORRECTION FOR INDEX CODING WITH SIDE INFORMATION 1525for all and all choices of .Wededuce that the rows of also satisfyBy Corollary 3.10, corresponds to a linear -IC. Therefore,by Lemma 3.5, part 2, has at least columns. We deducethatVI. RANDOM CODESIn this section, we prove an inexplicit upper bound on theoptimal length of the ECICs. The proof is based on constructinga random ECIC and analyzing its parameters.Theorem 6.1: Letof the ICSI problem. Then, there exists aof length ifdescribe an instance-ECIC overwhich concludes the proof.The following corollary from Proposition 4.6 and Theorem5.1 demonstrates that, for sufficiently large alphabets, a concatenationof a classical MDS error-correctingcodewithanoptimal(non-error-correcting) index code yields an optimal ECIC.However, as it was illustrated in Example 4.8, this does not holdfor the index coding schemes over small alphabets.Corollary 5.2 (MDS Error-Correcting Index Code): For,where(12)is the volume of the -ary sphere in .Proof: We construct a random matrix over ,row by row. Each row is selected independently of other rows,uniformly over .Define vector spacesfor all.Wealsodefine the following events:(11)Proof: From Theorem 5.1, we haveandOn the other hand, from Proposition 4.6for(by taking doubly extendedReed–Solomon (RS) codes). Therefore, for these , (11)holds.Remark 5.3: Let , ,andfor all .Let and .For ,let .Let .Then, is the (symmetric) odd cycle of length . Therefore,.From[7],.From -boundBy contrast, from Theorem 5.1The event represents the situation when the receivercannot recover . Then, by Corollary 3.9, the event isequivalent to . ThereforeFor a particular event ,(13)(14)There exists a matrix that corresponds to a -ECIC if. It is enough to require that the right-hand side of(13) is smaller than 1. By plugging in the expression in (14), weobtain a sufficient condition on the existence of a -ECICover :As there are no nontrivial binary MDS codes, we havefor all choices of . Therefore, for these choices, the-bound is at least as good as the Singleton bound.Remark 6.2: The bound in Theorem 6.1 does not take into accountthe structure of the sets ’s, other than their cardinalities.Therefore, this bound generally is weaker than the -bound. Onthe other hand, for a particular instance of the ICSI problem, it