On the Hamming distance of linear codes over a finite chain ring
G.H. Norton, Ana Sălăgean · IEEE Transactions on Information Theory · 2000
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~:~/spl alpha/~)~), where (C:/spl alpha/) is a submodule quotient, /spl alpha/ 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~:~/spl alpha/~)~ from the generator matrix/polynomials of C. We show that in general d(C)/spl les/d(C~) with equality for free codes (i.e., for free R-submodules of R/sup 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 characterize minimum distance separable (MDS) codes over R and prove several analogs of properties of MDS codes over finite fields. We compute the Hamming weight enumerator of a free MDS code over R.