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Many of the classic problems of coding theory are highly symmetric, whichmakes it easy to derive sphere-packing upper bounds and sphere-covering lowerbounds on the size of codes. We discuss the generalizations of sphere-packingand sphere-covering bounds to arbitrary error models. These generalizationsbecome especially important when the sizes of the error spheres are nonuniform.The best possible sphere-packing and sphere-covering bounds are solutions tolinear programs. We derive a series of bounds from approximations to packingand covering problems and study the relationships and trade-offs between them.We compare sphere-covering lower bounds with other graph theoretic lower boundssuch as Tur\'{a}n's theorem. We show how to obtain upper bounds by optimizingacross a family of channels that admit the same codes. We present ageneralization of the local degree bound of Kulkarni and Kiyavash and use it toimprove the best known upper bounds on the sizes of single deletion correctingcodes and single grain error correcting codes.

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