Nonlinear Recurrences and Arithmetic Codes

We show that the binary expansion of the code numbers in an arithmetic BN-code, where B = (2e − 1)/A and e is the order of 2 mod A, may be generated by a unique nonsingular shift register of order k = [log2 A]. The shift register is linear if and only if A = 2k − 1 or 2k−1 + 1. Besides the code numbers of the BN-code, this shift register will also generate the B′N-code where B′ = (2e′ − 1)/(2k − A) and the sequences (1) and (0) (the all-one and the all-zero sequences) and no other sequences. Nonlinear shift register are often used to generate long binary sequences. But one major problem is to predict the periods of the sequence generated, from the form of the shift register. This paper is a partial solution to this problem, since the periods of the sequences generated by the shift register above obviously are divisors of e or e′. (If D | A (D divides A) and eD is the order of 2 mod D then there exist φ(D) sequences of period eD, where φ(D) is the Euler φ function. A similar statement holds for a divisor D′ of 2k − A). The main problem in this connection is to synthesize a shift register which corresponds to a given A. This is easy when A = 2k −1 or 2k + 1. We solve this problem when A = 2k − 3 or 2k + 3. In principle there is no problem in doing this when A = 2k − 5, 2k + 5, 2k − 7, 2k + 7 and so forth, but it requires some work. The results we obtain may also be used for coding and decoding arithmetic codes.