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Singleton bounds for codes over finite rings. (English) Zbl 0979.94052

Let R be a finite commutative quasi-Froebenius (QF) ring and let V=R n be the free module of rank n consisting of all n-tuples of elements of R. A code C of length n over R is an R-submodule of V. For every x=(x 1 ,x 2 ,,x n )V and rR, the complete weight of x is defined by n r (x)=|{ix i =r}|. Let a 0 =0 and a r be a positive real number for any nonzero rR. A general weight function w(x) is defined by

w(x)= rR a r n r (x)·

For any linear [n,k,d] code over a finite field, the Singleton bound says that dn-k+1. The main result here is the following generalization of the Singleton bound for a general weight function over R.

Theorem. Let C be a code with minimum distance d over a finite commutative QF ring R. Let w(x) be a general weight function on C. Then

( d - 1 ) / An-log |R| |C|,

where A=max{a r rR} and [b] is the integer part of b.

Finally, the author presents some applications of his result to codes over l (and in particular 4 ) for the special weight functions corresponding to the Hamming, Lee and Euclidean weights.

MSC:
94B65Bounds on codes
16L60Quasi-Frobenius rings