# Embedding a θ-invariant code into a complete one

arXiv: https://arxiv.org/abs/1801.05164 · HTML: https://mlsanity.com/paper/1801.05164/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2018.

## Built on by (most cited first, 2 of 2)

- [Variable-Length Codes Independent or Closed with respect to Edit Relations](https://mlsanity.com/paper/2104.14185/) (2021)
- [When Variable-Length Codes Meet the Field of Error Detection](https://mlsanity.com/paper/2208.14681/) (2022)

## Abstract (arXiv)

Let A be a finite or countable alphabet and let θ 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 θ (θ-invariant for short) that is, languages L such that θ (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 θ-invariant codes. Moreover, we establish a formula which allows to embed any non-complete θ-invariant code into a complete one. As a consequence, in the family of the so-called thin θ--invariant codes, maximality and completeness are two equivalent notions.