Generator Matrices and Symmetrized Weight Enumerators of Linear Codes over Fpm + uFpm + vFpm + wFpm

Let u,v, and w be indeterminates over Fpm and let R=Fpm+uFpm+vFpm+wFpm, where p is a prime. Then, R is a ring of order p4m, and R≅Fpmu,v,wI with maximal ideal J=uFpm+vFpm+wFpm of order p3m and a residue field Fpm of order pm, where I is an appropriate ideal. In this article, the goal is to improve the understanding of linear codes over local non-chain rings. In particular, we investigate the symmetrized weight enumerators and generator matrices of linear codes of length N over R. In order to accomplish that, we first list all such rings up to the isomorphism for different values of the index of nilpotency l of J, 2≤l≤4. Furthermore, we fully describe the lattice of ideals of R and their orders. Next, for linear codes C over R, we compute the generator matrices and symmetrized weight enumerators, as shown by numerical examples.