On Constacyclic Codes Over A Kind Of Rings With Four Elements

Whether a code can be used in the practical application depends on the decoding of a Error-correcting Codes. Usually, Decoder are difficult to achieve compared with encoder, and much more complex for the codes that posses the strong ability of Error-correction. The decoder of cyclic codes can be divided as Error-correcting decoder and Error-detecting decoder for different purposes. The decoding speed of cyclic codes is very swift and cyclic codes has strong ability in Error-correction and Error-inspection, thus cyclic codes plays an important role in practice. People pay wide attention to the codes on the finite ring due to its special properties.A vast amount of articles and research has been done on error correcting codes over finite fields. It has been shown that certain good nonlinear binary codes such as binary Kerdock codes are the Gray images of some Z4-linear codes. Using the Gray map a new set of linear and nonlinear binary codes have been constructed as the Gray images of some codes over rings. So, this approach has helped to view some nonlinear binary codes as images of linear quaternary codes. Hence, the research on linear codes over rings with four elements has been popular. It is well known that there are four non-isomorphic rings with four elements; the finite field GF(4), the rinZ4, the ring Z2×Z2 and the ring R= Z2＋Z2={0,1,u,u+1} where u2=Omod2. In this paper, we focus on studying the cyclic codes on R. We obtain set of generators for these codes and the structure of their dual ideals and also obtain some basic properties of (1+u)—constacyclic codes over ring R.