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 $2 \cdot N/\log(N)$ 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.