Structures, Ranks and Minimal Distances of Cyclic Codes over Z p 2 + u Z p 2

Sami H. Saif ()
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Sami H. Saif: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Mathematics, 2025, vol. 13, issue 20, 1-18

Abstract: Let p be a prime and F p a finite field of order p . This paper investigates cyclic codes over the ring R p 2 , u = Z p 2 + u Z p 2 of order p 4 , where the nilpotent element u satisfies u 2 = 0 and p u ≠ 0 . The condition u 2 = 0 with p u ≠ 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 R p 2 , 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 R p 2 , u to F p 4 , we construct all but one of the ternary optimal codes of length 12 as images of cyclic codes over R 3 2 , u , with computations verified using the Magma system.

Keywords: rank; cyclic code; optimal code; minimal distance; finite ring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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