Generalized sphere packing bound: Basic principles

Kulkarni and Kiyavash recently introduced a new method to establish upper bounds on the size of deletion-correcting codes. This method is based upon tools from hypergraph theory. The deletion channel is represented by a hypergraph whose edges are the deletion balls (or spheres), so that a deletion-correcting code becomes a matching in this hypergraph. Consequently, a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph. Classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear-programming problem: the problem of finding fractional transversals. The method by Kulkarni and Kiyavash can be applied not only for the deletion channel but also for other channels, and in particular for those where the error spheres sizes are not all the same. This paper studies this method in its most general setup. We first show that if the error channel is regular and symmetric then the upper bound by this method coincides with the well-known sphere packing bound and thus is called here the generalized sphere packing bound. Even though this bound is explicitly given by a linear programming problem, finding its exact value may still be a challenging task. The art of finding the exact upper bound or slightly weaker ones is the assignment of weights to the hypergraph's vertices in a way that the satisfy the constraints in the linear programming problem. Every valid assignment yields an upper bound and the goal is to find assignments that provide strong upper bounds. We show that for graphs which satisfy a monotonicity property it is possible to find a general formula for such an assignment. Lastly, in order to simplify the complexity of the linear programming, we present a technique based upon graph automorphisms that in many cases can significantly reduce the number of variables and constraints in the linear programming problem. All of our results will be demonstrated and calculated for the Z channel which will be a case study in our work.