Analysis of cubic permutation polynomials for turbo codes

TL;DR

This paper investigates cubic permutation polynomials (CPPs) as interleavers in turbo codes, providing theoretical conditions, reducing search complexity, and demonstrating performance improvements over quadratic permutation polynomials (QPPs) for certain code lengths.

Contribution

It establishes necessary and sufficient conditions for CPPs to be null permutation polynomials and applies these to optimize interleaver design, improving turbo code performance.

Findings

CPP interleavers can outperform QPPs at some lengths.

Theoretical conditions reduce search complexity for CPPs.

Performance bounds show potential benefits of CPPs over QPPs.

Abstract

Quadratic permutation polynomials (QPPs) have been widely studied and used as interleavers in turbo codes. However, less attention has been given to cubic permutation polynomials (CPPs). This paper proves a theorem which states sufficient and necessary conditions for a cubic permutation polynomial to be a null permutation polynomial. The result is used to reduce the search complexity of CPP interleavers for short lengths (multiples of 8, between 40 and 352), by improving the distance spectrum over the set of polynomials with the largest spreading factor. The comparison with QPP interleavers is made in terms of search complexity and upper bounds of the bit error rate (BER) and frame error rate (FER) for AWGN and for independent fading Rayleigh channels. Cubic permutation polynomials leading to better performance than quadratic permutation polynomials are found for some lengths.