Analysis of Cubic Permutation Polynomials for Turbo Codes

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.

The limitation of permutation polynomial interleavers for turbo codes and a scheme for dithering permutation polynomials

“…In [2,7,8] few methods for finding good QPP or CPP based interleavers are given. These methods also take into account other parameters of QPP/CPP based interleavers (such as D parameter or spread and refined nonlinearity degree).…”

“…These methods also take into account other parameters of QPP/CPP based interleavers (such as D parameter or spread and refined nonlinearity degree). In Tables 6 and 7, the our dithered LPPs are compared with the QPPs from [2] or [7,8] and the CPPs from [2], for some lengths, considering the simulated FER and the minimum distance. It is observed that for all reported lengths dithered LPPs lead to higher or equal minimum distances (excepting CPP of length 40 and QPPs of lengths 320, 560, 848 and 1008, respectively).…”

“…For these lengths with exceptions, the 10 7 ×TUB(FER) values were computed, as shown in Tables 4 and 5. For lengths 448, 560 Table 6 Minimum distances (d min ), corresponding multiplicities of codewords (N d min ) and FER at high SNR for QPP from [2] or [8], CPP from [2] and dithered LPP interleavers for lengths N ≤ 200. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5].…”

“…Permutation polynomial (PP) based interleavers over integer rings have been widely studied [1][2][3][6][7][8]. In particular quadratic permutation polynomial (QPP) based interleavers were emphasized due to their simple implementation [3] as well as excellent performance [1].…”

“…Higher order PP based interleavers have also been investigated for better performance and implementation, in particular for cubic permutation polynomial (CPP) based interleavers [2,3]. However little is known for minimum distance of turbo codes with higher order PP based interleavers.…”