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

TL;DR

This paper extends classical sphere-packing and sphere-covering bounds to arbitrary and nonuniform error models in coding theory, using linear programming and graph-theoretic methods to derive tighter bounds and analyze trade-offs.

Contribution

It introduces generalized bounds for nonuniform error spheres, improves upper bounds on deletion and grain error correcting codes, and explores relationships between different bounds.

Findings

Derived bounds from linear programming approximations.

Compared sphere-covering bounds with Turán's theorem.

Improved upper bounds on deletion and grain error correcting codes.

Abstract

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