Analysis of cubic permutation polynomials for turbo codes

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.Comment: accepted for publication to Wireless Personal Communications (19 pages, 4 figures, 5 tables). The final publication is available at springerlink.co

When Is the Number of True Different Permutation Polynomials Equal to 0?

In this paper, we have obtained the prime factorization form of positive integers N for which the number of true different fourth- and fifth-degree permutation polynomials (PPs) modulo N is equal to zero. We have also obtained the prime factorization form of N so that the number of any degree PPs nonreducible at lower degree PPs, fulfilling Zhao and Fan (ZF) sufficient conditions, is equal to zero. Some conclusions are drawn comparing all fourth- and fifth-degree permutation polynomials with those fulfilling ZF sufficient conditions

On Dispersion and Nonlinearity Degree of QPP Interleavers

This paper shows a link between the dispersion and the nonlinearity degree of QPP (quadratic permutation polynomial) interleavers. An upper bound for the dispersion of QPP interleavers is derived. This upper bound is computed very simple, only depending on the coefficient of x2 of the polynomial and of the length of interleaver. The comparison with the real dispersion of QPP interleavers given by Takeshita in [1] leads to insignificant difference. Searching of QPP interleavers based on a metric including upper bound of dispersion and D parameter is equivalent with that based on metric

Capacity of Middleton Class-A Impulsive Noise Channel with Binary Input

In many applications, the Middleton Class-A model is used to describe the impulsive noise. A very useful and interesting aspect for a channel affected by such, non-Gaussian noise, is to find an expression for the channel capacity. In this paper we present the calculation of capacity for a channel affected by additive Middleton Class-A noise (AWCN), with binary input. We considered the case when the source is uniform, but also when it is not uniform. The channel capacities for impulsive noise, for various values of the parameters that describe its model, are compared with those for the additive white Gaussian noise (AWGN) channel. The numerical results showed that when the parameters A and T are close to 1, the capacity for impulsive noise channel is equal to that of Gaussian channel. When A and T decrease, the AWCN capacity grows. When the probability p0, the probability of bit 0, grows or when the encoding rate decreases, each channel capacity decreases. The Shannon limit values are also given for different encoding rates in the case of the two channels. We have shown that Signal-to-Noise Ratio (SNRb) in dB given only by the Gaussian component of AWCN is closer to SNRb in dB of AWGN, as the AWCN capacity increases

Lengths for Which Fourth Degree PP Interleavers Lead to Weaker Performances Compared to Quadratic and Cubic PP Interleavers

In this paper, we obtain upper bounds on the minimum distance for turbo codes using fourth degree permutation polynomial (4-PP) interleavers of a specific interleaver length and classical turbo codes of nominal 1/3 coding rate, with two recursive systematic convolutional component codes with generator matrix G = [ 1 , 15 / 13 ] . The interleaver lengths are of the form 16 Ψ or 48 Ψ , where Ψ is a product of different prime numbers greater than three. Some coefficient restrictions are applied when for a prime p i ∣ Ψ , condition 3 ∤ ( p i − 1 ) is fulfilled. Two upper bounds are obtained for different classes of 4-PP coefficients. For a 4-PP f 4 x 4 + f 3 x 3 + f 2 x 2 + f 1 x ( mod 16 k L Ψ ) , k L ∈ { 1 , 3 } , the upper bound of 28 is obtained when the coefficient f 3 of the equivalent 4-permutation polynomials (PPs) fulfills f 3 ∈ { 0 , 4 Ψ } or when f 3 ∈ { 2 Ψ , 6 Ψ } and f 2 ∈ { ( 4 k L − 1 ) · Ψ , ( 8 k L − 1 ) · Ψ } , k L ∈ { 1 , 3 } , for any values of the other coefficients. The upper bound of 36 is obtained when the coefficient f 3 of the equivalent 4-PPs fulfills f 3 ∈ { 2 Ψ , 6 Ψ } and f 2 ∈ { ( 2 k L − 1 ) · Ψ , ( 6 k L − 1 ) · Ψ } , k L ∈ { 1 , 3 } , for any values of the other coefficients. Thus, the task of finding out good 4-PP interleavers of the previous mentioned lengths is highly facilitated by this result because of the small range required for coefficients f 4 , f 3 and f 2 . It was also proven, by means of nonlinearity degree, that for the considered inteleaver lengths, cubic PPs and quadratic PPs with optimum minimum distances lead to better error rate performances compared to 4-PPs with optimum minimum distances