Radix- $2^{r}$ Arithmetic for Multiplication by a Constant

In this brief, radix-2
^r
arithmetic is explored to minimize the number of additions in the multiplication by a constant. We provide the formal proof that, for an N-bit constant, the maximum number of additions using radix- 2
^r
is lower than Dimitrov's estimated upper bound 2. N/log(N) using the double-base number system (DBNS). In comparison with the canonical signed digit (CSD) and the DBNS, the new radix- 2
^r
recoding requires an average of 23.12% and 3.07% less additions for a 64-bit constant, respectively.