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 [Formula: see text]. The interleaver lengths are of the form [Formula: see text] or [Formula: see text] , where [Formula: see text] is a product of different prime numbers greater than three. Some coefficient restrictions are applied when for a prime [Formula: see text] , condition [Formula: see text] is fulfilled. Two upper bounds are obtained for different classes of 4-PP coefficients. For a 4-PP [Formula: see text] , [Formula: see text] , the upper bound of 28 is obtained when the coefficient [Formula: see text] of the equivalent 4-permutation polynomials (PPs) fulfills [Formula: see text] or when [Formula: see text] and [Formula: see text] , [Formula: see text] , for any values of the other coefficients. The upper bound of 36 is obtained when the coefficient [Formula: see text] of the equivalent 4-PPs fulfills [Formula: see text] and [Formula: see text] , [Formula: see text] , 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 [Formula: see text] and [Formula: see text]. 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.