additive cyclic codes
The Z2Z4 additive codes has been introduced in [3] and intensively studied during last years. Recently, Z2Z4 additive cyclic codes has been defined in [1] and identified as Z4[x] modules of a certain ring. The duality of Z2Z4 additive cyclic codes has been studied in [5] In recent times, Z2Z4 additive codes were generalized to ZprZps additives codes in [2] They determine, in particular, the standard forms of generator and paritycheck matrices and present some bounds on the minimum distance. Let Zpr and Zps be the rings of integers modulo pr and ps, respectively, with p prime and r s. Since the residue field of Zpr and Zps is Zp, then an element b of Zpr could be written uniquely as b b0 pb1 pb2 pbr 1, and any element a Zps as a a0 pa1 pa2 pas 1, where bi,a j Zp. Then we can consider the surjective ring homomorphism p Zps Zpr where p(a) a mod pr. Note that p(pi) 0 if i> r. Let aZps and bZpr We define a multiplication as follows: a*b p(a)b. Then, Zpr is a Zps module with external multiplication given by p Since Zpr is commutative, then has the commutative property. Then, we can generalize this multiplication over the ring Zpr xZ b ps as follows. Let a be an element of Zps and u (u u' (u0,u1, ,ua 1 u0,u1, ,ub 1) Z a pr xZ b ps Then, a u (p(a)u0,p(a)u1, ,p(a)ua 1 au0,au1, ,aub 1) With this external operation the ring Zpr xZ b ps is also a Zps module.