Title: Linear codes over ℤ₄[x]/⟨x² + 2x⟩

Authors: Edgar Martínez-Moro; Steve Szabo; Bahattin Yildiz

Addresses: Institute of Mathematics, Applied Mathematics Department, University of Valladolid, Valladolid, Spain ' Department of Mathematics and Statistics, Eastern Kentucky University, 521 Lancaster Avenue, Richmond, KY 40475, USA ' Department of Mathematics, Fatih University, 34500 Istanbul, Turkey

Abstract: In this work codes over one of seven local Frobenius non-chain rings of order 16 are studied. The ring structure is discussed showing both the similarities and differences to a previously studied ring. A duality preserving Gray map is given and is used to present MacWilliams identities and self-dual codes. Connections between these self-dual codes and real unimodular lattices are also discussed. Some extremal Type II ℤ₄-codes are provided as images of codes over this ring. ℤ₄-codes that are images of linear codes over the studied ring are characterised through automorphism groups and some well-known families of ℤ₄-codes (different versions of quaternary Reed-Muller codes) are proved to be linear over it.

Keywords: Frobenius ring; chain ring; local Frobenius non-chain rings; codes over rings; Reed-Muller codes; ℤ4 codes; self-dual codes; ring structure; linear codes.

DOI: 10.1504/IJICOT.2015.068698

International Journal of Information and Coding Theory, 2015 Vol.3 No.1, pp.78 - 96

Received: 14 Jan 2015
Accepted: 08 Feb 2015
Published online: 09 Apr 2015 *

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