Minimum Distance of Turbo Codes with Permutation Polynomial-Based Interleavers

Abstract

In this chapter we present upper bounds on the minimum distance of turbo codes using QPP interleavers obtained by Eirik Rosnes in (2012), as well as an upper bound on the minimum distance of turbo codes using PP-based interleaver of any degree given in Ryu et al. (2015). We note that most of these upper bounds are partial, meaning that they are valid only for certain interleaver lengths complying with certain constraints on their prime decomposition. A single upper bound is general (see Theorem 5.10, Eq. (5.112)), meaning that it holds for any interleaver length, but there is a constraint on the minimum degree of the inverse polynomials to be equal to two (i.e. the QPP interleaver must have an inverse QPP (Ryu and Takeshita 2006)). Because some of the above mentioned upper bounds use inverse polynomials of QPPs, in the first section of this chapter we present the necessary and sufficient condition for a QPP to admit at least one quadratic inverse, derived by Jonghoon Ryu and Oscar Y. Takeshita in Ryu and Takeshita (2006). At the end of the first section we give, without proof, a theorem from Ryu (2007), Ryu and Takeshita (2011) which allows us to determine all inverse permutation polynomials of a QPP of the smallest degree.