Generalized sphere packing bound: Applications

Abstract

In this paper we study a generalization of the sphere packing bound for channels that are not regular (the size of balls with a fixed radius is not necessarily the same). Our motivation to tackle this problem is originated by a recent work by Kulkarni and Kiyavash who introduced a method, based upon tools from hypergraph theory, to calculate explicit upper bounds on the cardinalities of deletion-correcting codes. Under their setup, the deletion channel is represented by a hypergraph such that every deletion ball is a hyperedge. Since every code is a matching in the hypergraph, an upper bound on the codes is given by an upper bound on the largest matching in a hypergraph. This bound, called here the generalized sphere packing bound, can be found by the solution of a linear programming problem. We similarly study and analyze specific examples of error channels. We start with the Z channel and show how to exactly find the generalized sphere packing bound for this setup. Next studied is the non-binary limited magnitude channel both for symmetric and asymmetric errors. We focus on the case of single error and derive upper bounds on the generalized sphere packing bound in this channel. We follow up on the deletion case, which was the original motivation of the work by Kulkarni and Kiyavash, and show how to improve upon their upper bounds for the single deletion case. Finally, we apply this method for projective spaces and find its generalized sphere packing bound for the single-error case.