**On the distinctness of binary sequences derived from primitive sequences modulo square-free odd integers**

*Qun-Xiong Zheng, Wen-Feng Qi and Tian Tian*

**Abstract: **Let M be a square-free odd integer and Z/(M) the integer residue ring modulo M. This paper studies the distinctness of primitive sequences over Z/(M) modulo 2. Recently, for the case of M = pq, a product of two distinct prime numbers p and q, the problem has been almost completely solved. As for the case that M is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order 2k+1 over Z/(M) is distinct modulo 2. Besides as an independent interest, the paper also involves two distribution properties of primitive sequences over Z/(M), which related closely to our main results.

**Category / Keywords: **foundations / integer residue rings, linear recurring sequences, primitive polynomials, primitive sequences, modular reduction

**Date: **received 3 Jan 2012

**Contact author: **qunxiong_zheng at 163 com

**Available format(s): **PDF | BibTeX Citation

**Note: **The manuscript was summitted to IEEE Transactions on Information Theory in Aug. 2011.

**Version: **20120105:055938 (All versions of this report)

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