Additive Negacyclic Z₂Z₄-Codes

Since the 1950’s, binary linear codes have been considered to be one of important research fields of error-correcting codes theory. While the nonlinear codes have higher information rate than linear codes, there have been lots of difficulties in encoding and decoding because of their complicated structure. Nechaev first began to study quaternary sequences in 1991 and then Hammons etal proved the quaternary linear codes that they constructed can be seen as some classical binary nonlinear codes under Gray mapping. It transforms the study of nature of binary nonlinear codes into quaternary linear codes so that scholars focus on studying error-correcting codes over finite rings. Recently, a new class of error-correcting codes called additive Z2Z4-codes emerged that generalizes binary linear codes and quarternary linear codes. It provides a new idea and direction to the research of coding theory.Cyclic code is one of important subclasses of linear codes. Lots of important codes such as Golay codes, Hamming codes, BCH codes and so on can be transformed into cyclic codes or just belong to. We can find a variety of practical decoding methods since the rich algebraic structure of cyclic codes can be analyzed and constructed by algebraic methods. Negacyclic codes have important research value either in theory or in practical applications as some of their excellent properties are similar to cyclic codes. In the late 1960s, Berlekamp first put forward the concept of negacyclic codes over finite fields and then Wolfmann studied negacyclic Z4-codes of odd lengths and gave many important properties of these codes. Since then, the research of negacyclic codes over finite rings has aroused the interest of scholars so that the further algebraic properties and applications of negacyclic codes have been exploring in more fields.In this paper,we mainly study additive negacycle Z2Z4-codes. Firstly, we define the additive negacycle Z2Z4-codes and give a set of generator polynomials for this codes as a Z4[x]-submodules. Secondly, we find a set of generators for negacycle Z2Z4-codes as Z4-submodules, give the calculating formula of the number of the codewords and obtain a number of binary linear codes with optimal parameters from the additive negacycle Z2Z4-codes which can be transform into binary linear codes by Gray mapping. Finally, we discuss maximum distance separable additive negacycle Z2Z4-codes and give an infinite families of this kind of codes; We also define the additive reversible negacycle Z2Z4-codes and discuss their structure.