Embedding rationally independent languages into maximal ones

TL;DR

This paper introduces a method for embedding a given language into a maximal one while preserving a rational independence property, applicable to various coding theory properties and often computationally feasible.

Contribution

It extends existing embedding techniques to order-decreasing rational relations, broadening applicability to multiple coding properties and enabling effective computation.

Findings

Method works for many known coding properties.

Applicable to both noiseless and noisy coding domains.

Maximal embeddings are often effectively computable.

Abstract

We consider the embedding problem in coding theory: given an independence (a code-related property) and an independent language $L$, find a maximal independent language containing $L$. We consider the case where the code-related property is defined via a rational binary relation that is decreasing with respect to any fixed total order on the set of words. Our method works by iterating a max-min operator that has been used before for the embedding problem for properties defined by length-increasing-and-transitive binary relations. By going to order-decreasing rational relations, represented by input-decreasing transducers, we are able to include many known properties from both the noiseless and noisy domains of coding theory, as well as any combination of such properties. Moreover, in many cases the desired maximal embedding is effectively computable.