Chapter4 2 4. Z2Z2[u]-Linear and Z2Z2[u]-Cyclic Codes

Abstract Additive codes were first introduced by Delsarte in 1973 as subgroups of the underlying abelian group in a translation association scheme. In the case where the association scheme is the Hamming scheme, that is, when the underlying abelian group is of order 2 n , the additive codes are of the form Z 2 α × Z 4 β with α + 2 β = n . In 2010, Borges et al. introduced Z 2 Z 4 -additive codes which they defined them as the subgroups of Z 2 α × Z 4 β . In this chapter we introduce Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes where Z 2 = 0 1 is the binary field and Z 2 u = 0 1 u 1 + u is the ring with four elements and u 2 = 0 . We give the standard forms of the generator and parity-check matrices of Z 2 Z 2[u]-linear codes. Further, we determine the generator polynomials for Z 2 Z 2 u -linear cyclic codes. We also present some examples of Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes.