Codes over Frobenius Ring T

This paper focuses on the construction and analysis of codes over the Frobenius ring T. By exploiting the idempotent structure of the ring, codes over T are decomposed into direct sums of component codes over a related local Frobenius non-chain ring. A gray map is constructed and shown to be bijective and distance-preserving, allowing code parameters to be determined from their gray images. Linear codes and cyclic codes over the considered ring are completely characterized, and explicit descriptions of their generating sets and cardinalities are obtained. The duals of cyclic codes are also examined, and the gray map is shown to preserve orthogonality and self-duality. Well-parameterized codes were obtained over the ring T.