Abstract
Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, (1 + u)-constacyclic codes over Z4 + uZ4 of any length are studied. A new Gray map between Z4 + uZ4 and Z44 is defined. By means of this map, it is shown that the Z4 Gray image of a (1 + u)-constacyclic code of length n over Z4 + uZ4 is a cyclic code over Z4 of length 4n. Furthermore, by combining the classical Gray map between Z4 and F22, it is shown that the binary image of a (1 + u)-constacyclic code of length n over Z4 + uZ4 is a distance invariant binary quasi-cyclic code of index 4 and length 8n. Examples of good binary codes are constructed to illustrate the application of this class of codes.
Highlights
Several new classes of rings have been studied in connection with coding theory
The structures of cyclic codes and (1 + u)-constacyclic codes over F2 + uF2 + vF2 + uvF2 were studied and many optimal binary linear codes were constructed from such codes in Yildiz and Karadenniz (2011a, b)
Motivated by the works in Yildiz and Aydin (2014) and Yildiz and Karadenniz (2014), we focus on constacyclic codes over Z4 + uZ4 and intend to construct good binary codes from such codes
Summary
Several new classes of rings have been studied in connection with coding theory. In Yildiz and Karadenniz (2010a, b), the authors introduced the ring F2 + uF2 + vF2 + uvF2 and discussed linear and self-dual codes over F2 + uF2 + vF2 + uvF2. The structures of cyclic codes and (1 + u)-constacyclic codes over F2 + uF2 + vF2 + uvF2 were studied and many optimal binary linear codes were constructed from such codes in Yildiz and Karadenniz (2011a, b). In Kai et al (2012), the authors introduced a composite Gray map from F2 + uF2 + vF2 + uvF2 to F24 and proved that the image of a (1 + u)-constacyclic code of length n over F2 + uF2 + vF2 + uvF2 under the Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n.
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