Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective

TL;DR

This paper introduces algebraic-geometric methods to design permutation polynomial interleavers with maximum spread, proposes a new combined metric for interleaver quality, and demonstrates improved turbo code performance through simulations.

Contribution

It presents an infinite sequence of quadratic permutation polynomials for maximum-spread interleavers and introduces a novel non-linearity metric combined with spread factor for better interleaver evaluation.

Findings

Maximum-spread interleavers achieve the upper bound of spread factor.

The new metric correlates well with turbo code performance.

Simulation results show significant frame error rate improvements.

Abstract

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are important because they admit analytical designs and simple, practical hardware implementation. The spread factor of an interleaver is a common measure for turbo coding applications. Maximum-spread interleavers are interleavers whose spread factors achieve the upper bound. An infinite sequence of quadratic permutation polynomials over integer rings that generate maximum-spread interleavers is presented. New properties of permutation polynomial interleavers are investigated from an algebraic-geometric perspective resulting in a new non-linearity metric for interleavers. A new interleaver metric that is a function of both the non-linearity metric and the spread factor is proposed. It is numerically demonstrated that the spread factor has a diminishing importance with the block length. A table of good interleavers for a variety of interleaver lengths according to the new metric is listed. Extensive computer simulation results with impressive frame error rates confirm the efficacy of the new metric. Further, when tail-biting constituent codes are used, the resulting turbo codes are quasi-cyclic.