Homogeneous Weights of Linear Codes over 𝔽pm[u, v] of Order p4m

Let R=Fpm[u,v] be the ring whose order is p4m, where p denotes a prime number. The algebraic structure of R is substantially influenced by the relationships among the elements u2, uv, and v2. We establish that R is formed over Fpm when p≠2 and u2=γv2, where γ is a non-zero element of Fpm and uv=0. For p=2, the situations are those where u2=v2=0 and uv≠0. After that, we discuss how to make matrices that are linked to the symmetrical weight enumerators of linear codes over these rings. These matrices are essential in the examination of coding theory over Frobenius rings. Utilizing these matrices makes it possible to determine the homogeneous distances of linear codes over the ring R. Our results show a connection between the uniform distances of linear codes and the matrices of symmetrized weight enumerators. This helps us understand the behavior and characteristics of these codes.