Codeword Distinguishability in Minimum Diversity Decoding

We re-take a coding-theoretic notion which goes back to Cl. Shannon: codeword distinguishability. This notion is standard in zero-error information theory, but its bearing is definitely wider and it may help to better understand new forms of coding, as we argue below. In our approach, the underlying decoding principle is very simple and very general: one decodes by trying to minimise the diversity (in the simplest case the Hamming distance) between a codeword and the output sequence observed at the end of the noisy transmission channel. Symmetrically and equivalently, minimum-diversity decoders and codeword distinguishabilities may be replaced by maximum-similarity decoders and codeword confusabilities. The operational meaning of codeword distinguishability is made clear by a reliability criterion, which generalises the well-known criterion on minimum Hamming distances for error-correction codes. We investigate the formal properties of distinguishabilities versus diversities; these two notions are deeply related, and yet essentially different. An encoding theorem is put forward, which supports and suggests old and new code constructions. In a list of case studies, we examine channels with crossovers and erasures, or with crossovers, deletions and insertions, a channel of cryptographic interest, and the case of a few “odd distances” taken from DNA word design. ∗A. Sgarro is with the Department of Mathematics and Computer Science, University of Trieste, 34100 Trieste, Italy. E-mail: [email protected] †L. Bortolussi is with the Department of Mathematics and Computer Science, University of Udine, 33100 Udine, Italy. E-mail: [email protected] ‡Partly supported by GNCS, INdAM.