The number of different true permutation polynomial based interleavers under Zhao and Fan sufficient conditions

Abstract

Interleavers are important blocks of turbo codes, their type and dimensions having a significant influence on the performances of the mentioned codes. If appropriately chosen, permutation polynomial (PP) based interleavers lead to remarkable performances of these codes. Necessary and sufficient conditions for a polynomial to be PP are known only for degrees up to three. Sufficient conditions for the coefficients of a polynomial of any degree to be PP were presented by Zhao and Fan (Electron Lett 43(8):449---451, 2007). In this paper, we determine the number of true different PPs of degrees 3, 4 and 5 that meet the sufficient conditions given by Zhao and Fan. This is of particular interest when we need to find higher than second degree PP based interleavers for turbo codes, because the Zhao and Fan sufficient conditions avoid brute force searching of the PP coefficients. Some remarks are made for the Long Term Evolution standard interleaver lengths.