Lattice Codes for the Binary Deletion Channel

TL;DR

This paper introduces a novel lattice-based approach to constructing deletion codes for the binary deletion channel, leveraging lattice theory and coding over integers to improve code design.

Contribution

It presents a new method for creating deletion codes by reducing the problem to lattice construction via Construction A and run length coding.

Findings

Lower bounds on code sizes for the Manhattan metric

Use of generalized theta series to analyze lattice properties

Reduction of deletion code construction to lattice expurgation

Abstract

The construction of deletion codes for the Levenshtein metric is reduced to the construction of codes over the integers for the Manhattan metric by run length coding. The latter codes are constructed by expurgation of translates of lattices. These lattices, in turn, are obtained from Construction~A applied to binary codes and $\Z_4-$codes. A lower bound on the size of our codes for the Manhattan distance are obtained through generalized theta series of the corresponding lattices.