Channels with Synchronization/Substitution Errors and Computation of Error Control Codes

TL;DR

This paper introduces a probabilistic method for constructing near-maximal error-detecting and error-correcting codes for channels with substitution and synchronization errors, using a transducer-based error model.

Contribution

It presents a randomized algorithm for generating error-detecting codes close to maximal, modeled with transducers for complex error types, and demonstrates its implementation and testing.

Findings

Algorithm successfully generates near-maximal codes

Transducer model effectively captures complex errors

Implementation shows promising results on various channels

Abstract

We introduce the concept of an \ff-maximal error-detecting block code, for some parameter \ff{} between 0 and 1, in order to formalize the situation where a block code is close to maximal with respect to being error-detecting. Our motivation for this is that constructing a maximal error-detecting code is a computationally hard problem. We present a randomized algorithm that takes as input two positive integers $N,\ell$, a probability value \ff, and a specification of the errors permitted in some application, and generates an error-detecting, or error-correcting, block code having up to $N$ codewords of length $\ell$. If the algorithm finds less than $N$ codewords, then those codewords constitute a code that is \ff-maximal with high probability. The error specification (also called channel) is modelled as a transducer, which allows one to model any rational combination of substitution and synchronization errors. We also present some elements of our implementation of various error-detecting properties and their associated methods. Then, we show several tests of the implemented randomized algorithm on various channels. A methodological contribution is the presentation of how various desirable error combinations can be expressed formally and processed algorithmically.