A class of linear codes of length 2 over finite chain rings

Let F-pm be a finite field of cardinality p(m), where p is an odd prime, k, lambda be positive integers satisfying lambda >= 2, and denote K= Fp(m) [x]/ < f (x)(lambda pk)>, where f (x) is an irreducible polynomial in F-pm [a]. In this note, for any fixed invertible element omega is an element of K-x, we present all distinct linear codes S over K of length 2 satisfying the condition: (omega f (x)p(k) a(1), a(0)) is an element of S for all (a(0), a(1)) is an element of S. This conclusion can be used to determine the structure of (delta + alpha u(2))-constacyclic codes over the finite chain ring F-pm [u]/< u(2 lambda)> of length np(k) for any positive integer n satisfying gcd(p, n) = 1.