Permutation Polynomial Based Interleavers. Conditions on Coefficients

Abstract

In this chapter we obtain the conditions on the coefficients of a polynomial so that it is a PP modulo a positive integer. Definition of a PP-based interleaver is given in Sect. 3.1. Conditions on the coefficients of a polynomial of arbitrary degree so that it is a PP modulo a power of 2 are given in Sect. 3.2. Necessary and sufficient conditions so that a polynomial is a PP modulo a power of an arbitrary prime are given in Sect. 3.3. Simplified necessary and sufficient conditions on the coefficients of a polynomial so that it is a PP modulo an arbitrary positive integer are given in Sect. 3.4. Conditions on the coefficients of a polynomial of first, second, third, fourth and fifth degree so that it is a LPP, QPP, CPP, permutation polynomial of fourth degree (4-PP), and permutation polynomial of fifth degree (5-PP), are obtained in Sects. 3.5, 3.6, 3.7, 3.8, and 3.9, respectively. Zhao and Fan sufficient conditions (Zhao and Fan 2007) on the coefficients of a polynomial of arbitrary degree so that is a PP are given in Sect. 3.10. Finally, an algorithm obtained by Weng and Dong (2008) to get all PPs up to five degree is presented in Sect. 3.11.