Minimal weight expansions in Pisot bases

Abstract. For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base 2. In this paper, we consider numeration systems with respect to real bases β which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When β is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits ±1 and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form.