Non-asymptotic Upper Bounds for Deletion Correcting Codes

TL;DR

This paper derives explicit non-asymptotic upper bounds on the sizes of deletion correcting codes, improving existing bounds and modeling the problem via hypergraph matchings and linear programming.

Contribution

It introduces new non-asymptotic bounds for deletion correcting codes, including for constrained sources, and models the problem as a hypergraph matching and linear program.

Findings

Upper bounds match known asymptotic bounds of Levenshtein and Tenengolts.

Improved bounds on the asymptotic rate function.

Support for the conjecture that Varshamov-Tenengolts codes are optimal in the binary case.

Abstract

Explicit non-asymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for $q$-ary alphabet and string length $n$ is shown to be of size at most $\frac{q^n-q}{(q-1)(n-1)}$. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The non-asymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known non-asymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.