Undid revision 1049189412 by Barkercoder (talk): I don't see how Barker codes meet the definition of a linear code; or perhaps I misunderstand, but there's no explanation/references
[[Hamming space]]s over non-field alphabets have also been considered, especially over [[finite ring]]s, (most notably [[Galois ring]]s over [[modular arithmetic|'''Z'''<sub>4</sub>]]). This givinggives rise to [[module (mathematics)|module]]s instead of vector spaces and [[ring-linear code]]s (identified with [[submodule]]s) instead of linear codes. The typical metric used in this case the [[Lee distance]]. There exist a [[Gray isometry]] between <math>\mathbb{Z}_2^{2m}</math> (i.e. GF(2<sup>2m</sup>)) with the Hamming distance and <math>\mathbb{Z}_4^m</math> (also denoted as GR(4,m)) with the Lee distance; its main attraction is that it establishes a correspondence between some "good" codes that are not linear over <math>\mathbb{Z}_2^{2m}</math> as images of ring-linear codes from <math>\mathbb{Z}_4^m</math>.<ref name="Greferath2009">{{cite book |editor=Massimiliano Sala |editor2=Teo Mora |editor3=Ludovic Perret |editor4=Shojiro Sakata |editor5=Carlo Traverso|title=Gröbner Bases, Coding, and Cryptography|year=2009|publisher=Springer Science & Business Media|isbn=978-3-540-93806-4|chapter=An Introduction to Ring-Linear Coding Theory|author=Marcus Greferath}}</ref><ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Main_Page|title=Encyclopedia of Mathematics|website=www.encyclopediaofmath.org}}</ref><ref name="Lint1999">{{cite book|author=J.H. van Lint|title=Introduction to Coding Theory|url=https://archive.org/details/introductiontoco0000lint_a3b9|url-access=registration|year=1999|publisher=Springer|isbn=978-3-540-64133-9|edition=3rd|at=Chapter 8: Codes over ℤ<sub>4</sub>}}</ref>
More recently,{{when|date=May 2015}} some authors have referred to such codes over rings simply as linear codes as well.<ref name="DoughertyFacchini2015">{{cite book |editor=Steven Dougherty |editor2=Alberto Facchini |editor3=Andre Gerard Leroy |editor4=Edmund Puczylowski |editor5=Patrick Sole|title=Noncommutative Rings and Their Applications|chapter-url=https://books.google.com/books?id=psrXBgAAQBAJ&pg=PA80|year=2015|publisher=American Mathematical Soc.|isbn=978-1-4704-1032-2|page=80|chapter=Open Problems in Coding Theory|author=S.T. Dougherty |author2=J.-L. Kim |author3=P. Sole}}</ref>
