We generalise structure theorems of Calderbank and Sloane for linear and cyclic codes over ‫ޚ‬ p a to a finite chain ring. Our results are more detailed and do not use non-trivial results from Commutative Algebra.

Codes C 1 , . . . , C M of length n over F q and an M × N matrix A over F q define a matrix-product codeWe study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ⊥ . It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be 'non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary 'Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ⊥ can be described using C ⊥ 1 , . . . , C ⊥ M and a transformation of A. This yields d(C ⊥ ). Finally we show that an NSC matrix-product code is a generalized concatenated code.

Let R be a finite chain ring (e.g. a Galois ring), K its residue field and C a linear code over R. We prove that d(C), the Hamming distance of C, is d((C : α)), where (C : α) is a submodule quotient, α is a certain element of R and denotes the canonical projection to K. These two codes also have the same set of minimal codeword supports. We explicitly construct a generator matrix/polynomial of (C : α) from the generator matrix/polynomials of C. We show that in general d(C) ≤ d(C) with equality for free codes (i.e. for free Rsubmodules of R n) and in particular for Hensel lifts of cyclic codes over K. Most of the codes over rings described in the literature fall into this class. We characterise MDS codes over R and prove several analogues of properties of MDS codes over finite fields. We compute the Hamming weight enumerator of a free MDS code over R.

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