# On the Hamming distance of linear codes over a finite chain ring

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 R-
submodules of Rn) 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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## History

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- Science

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- Computer Science