Let be a finite commutative quasi-Froebenius (QF) ring and let be the free module of rank consisting of all -tuples of elements of . A code of length over is an -submodule of . For every and , the complete weight of is defined by . Let and be a positive real number for any nonzero . A general weight function is defined by
For any linear code over a finite field, the Singleton bound says that . The main result here is the following generalization of the Singleton bound for a general weight function over .
Theorem. Let be a code with minimum distance over a finite commutative QF ring . Let be a general weight function on . Then
where and is the integer part of .
Finally, the author presents some applications of his result to codes over (and in particular for the special weight functions corresponding to the Hamming, Lee and Euclidean weights.