Embedding a $\theta$-invariant code into a complete one

TL;DR

This paper explores the properties of $ heta$-invariant codes, extending the defect theorem, and provides methods to embed non-complete codes into complete ones, revealing the equivalence of maximality and completeness in certain cases.

Contribution

It extends the defect theorem for $ heta$-invariant codes and introduces a formula to embed any non-complete $ heta$-invariant code into a complete one.

Findings

Established an extension of the defect theorem for $ heta$-invariant codes.

Provided examples of finite complete $ heta$-invariant codes.

Derived a formula for embedding non-complete $ heta$-invariant codes into complete codes.

Abstract

Let A be a finite or countable alphabet and let $\theta$ be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short) that is, languages L such that $\theta$ (L) is a subset of L.We establish an extension of the famous defect theorem. With regards to the so-called notion of completeness, we provide a series of examples of finite complete $\theta$-invariant codes. Moreover, we establish a formula which allows to embed any non-complete $\theta$-invariant code into a complete one. As a consequence, in the family of the so-called thin $\theta$--invariant codes, maximality and completeness are two equivalent notions.