Error Correction for Index Coding With Side Information

Abstract

Abstract-A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998).In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Errorcorrecting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols δ, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortestlength error-correcting code with a prescribed error-correcting capability in the classical coding theory.The Singleton bound and two other bounds, referred to as the α-bound and the κ-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC.Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC.Finally, the notion of static ECIC, which is suitable for use with a family of instances of an index coding problem, is introduced. Several bounds on the length of static ECIC's are derived, and constructions for static ECIC's are discussed. Connections of these codes to weakly resilient Boolean functions are established.

An Optimal Linear Error Correcting Delivery Scheme for Coded Caching with Shared Caches

“…Vertex i represents the message x i and each hyperedge represents a receiver. In [14], the min-rank of a hypergraph H over F q is defined as,…”

“…Lower and upper bounds on N q [H, δ] were established in [14]. The Lower bound is known as the α-bound and the upper bound is known as the κ-bound.…”

“…Since α and κ bounds meet for I(M SC , d worst ), the optimal linear error correcting delivery scheme here would be concatenation of SC delivery scheme with an optimal classical error correcting delivery scheme which corrects δ errors. Decoding can be done by syndrome decoding for error correcting index codes proposed in [14]. We give an example for which we construct optimal error correcting delivery scheme for coded caching problems with SC prefetching.…”

Optimal Scalar Linear Index Codes for One-Sided Neighboring Side-Information Problems

“…Let E ⊆ [0 : K − 1]. We denote u ⊳ E whenever supp(u) ⊆ E. Then, the minrank q (G) over F q is defined [11] as min{rank Fq…”

Dynamic index coding gain over a complete bi-directional side information graph

“…In the second side, the generalization of the conventional index coding problem scenario is considered [1], [15], [16]. In our paper, which fits in this direction, we consider the problem of dynamic index coding introduced in [1] which is a more practical and complex version of the index coding problem dealing with random traffics instead of a fixed batch of packets.…”