  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.
