On Linear Codes over Local Rings of Order p 4

Suppose R is a local ring with invariants p , n , r , m , k and m r = 4 , that is R of order p 4 . Then, R = R 0 + u R 0 + v R 0 + w R 0 has maximal ideal J = u R 0 + v R 0 + w R 0 of order p ( m − 1 ) r and a residue field F of order p r , where R 0 = G R ( p n , r ) is the coefficient subring of R . In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R . In order to accomplish that, we first list all such rings up to isomorphism for different values of p , n , r , m , k . Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R , we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them.