In this paper, it is shown that some results in the theory of rank-metric codes over finite fields can be extended to finite commutative principal ideal rings. More precisely, the rank metric is generalized and the rank-metric Singleton bound is established. The definition of Gabidulin codes is extended and it is shown that its properties are preserved. The theory of Gröbner bases is used to give the unique decoding, minimal list decoding, and error-erasure decoding algorithms of interleaved Gabidulin codes. These results are then applied in space-time codes and in random linear network coding as in the case of finite fields. Specifically, two existing encoding schemes of random linear network coding are combined to improve the error correction.

This paper shows that q-ary images of some cyclic codes can be put into concatenated form in the sense of Jensen (1985). This description utilizes an isomorphism between the GF(q)-algebras GF(q) [z] / (znm- 1) and GF (q )[x, y] / (xn- 1, ym- x). We also describe an element of the constacyclic outer code, which is of great help in determining one of its generator polynomials.

A characterization is given of systematic codes over $GF( q )$ that are q-ary images of linear codes over $GF( q^2 )$.