On the number of optimal linear index codes for unicast index coding problems

Abstract

Abstract-An index coding problem arises when there is a single source with a number of messages and multiple receivers each wanting a subset of messages and knowing a different set of messages a priori. The noiseless Index Coding Problem is to identify the minimum number of transmissions (optimal length) to be made by the source through noiseless channels so that all receivers can decode their wanted messages using the transmitted symbols and their respective prior information. Recently [8], it is shown that different optimal length codes perform differently in a noisy channel. Towards identifying the best optimal length index code one needs to know the number of optimal length index codes. In this paper we present results on the number of optimal length index codes making use of the representation of an index coding problem by an equivalent network code. Our formulation results in matrices of smaller sizes compared to the approach of Kotter and Medard [6]. Our formulation leads to a lower bound on the minimum number of optimal length codes possible for all unicast index coding problems [1] which is met with equality for several special cases of the unicast index coding problem. A method to identify the optimal length codes which lead to minimum-maximum probability of error is also presented.

Unicast-Uniprior Index Coding Problems: Minrank and Criticality

“…However, for some very specific problem settings minrank can be computed in polynomial time [7], [10]. In [10], the notion of critical graphs in index coding was introduced and it was stated that certain side information bits can be called critical, if removing the corresponding edges in the graph G strictly reduces the capacity region (see Section II-A) of the index coding problem.…”

“…Step 3 of the algorithm computes minrank of single unicast uniprior problems, using the method in [7], which prunes the corresponding sideinformation graph to obtain the maximum number of disjoint unicycles.…”

“…x10 x8 3) The three single-unicast uniprior problems that are obtained from the above table are as follows: 4) We evaluate the minranks of the three problems using the method given in [7]. rk 11 = 4, rk 12 = 4, rk 13 = 1.…”

“…The wanted message is the table entry and the side-information is same as the side-information derived from the unicast-uniprior problem. The minrank of a single unicast uniprior index coding problem can be computed in linear time as discussed in [7]. 4) Let the minranks along each column in T i be denoted by rk i1 , rk i2 , .…”

“…One unicycle is equivalent to one minimally dependent set of rows and hence reduces the rank by one. Step 3 of the algorithm computes minrank of single unicast uniprior problems, using the method in [7], which prunes the corresponding sideinformation graph to obtain the maximum number of disjoint unicycles.…”