Uniqueness of Weighted Code Representations

Decimal computers ordinarily use a binary-coded decimal representation. One class of binary-coded decimal digits is the so-called four-bit weighted code representation with weights ¿1 ¿2 ¿3 ¿4. Each ¿i is a nonzero integer in the range ¿9¿¿i¿9, and the set of weights must have the property that every decimal digit can be represented by the sum ¿i=14 bi ¿i, with the bi being 0 or 1. For some weighted codes the weights are such that some digits can be represented by more than one sum of the specified form. For example, the 7421 weighted code has the property that 7 may be represented either as 1000 or as 0111. This paper produces a necessary and sufficient condition on the weights of a weighted code for the unique representation of each digit by a sum of the specified form. Further, all possible sets of weights are displayed.