Generalized sphere-packing upper bounds on the size of codes for combinatorial channels

A code for a combinatorial channel is a feasible point in an integer linear program derived from that channel. Sphere-packing upper bounds are closely related to the fractional relaxation of this program. When bounding highly symmetric channels, this formulation can often be avoided, but it is essential in less symmetric cases. We present a few low-complexity upper bounds on the value of the relaxed linear program. We also discuss a more general bound derived from the codeword constraint graph for the channel. This bound is not necessarily computationally tractable. When there is a family of channels with the same constraint graph, tractable bounds can be applied to each channel and the best bound will apply to the whole family.