Structures, Ranks and Minimal Distances of Cyclic Codes over Zp2uZp2+

Let p be a prime and Fp a finite field of order p. This paper investigates cyclic codes over the ring Rp2,u=Zp2+uZp2 of order p4, where the nilpotent element u satisfies u2=0 and pu≠0. The condition u2=0 with pu≠0 is crucial, as it creates a nontrivial interaction between the components of the ring, allowing the construction of new codes with enhanced structural and distance properties. We provide explicit generating sets for cyclic codes over Rp2,u and study fundamental parameters such as their rank and Hamming distance. In the case gcd(n,p)=1, we show that cyclic codes can be generated by just two polynomials, which allows a complete determination of their rank and minimal Hamming distance distributions. Furthermore, using the Gray map from Rp2,u to Fp4, we construct all but one of the ternary optimal codes of length 12 as images of cyclic codes over R32,u, with computations verified using the Magma system.