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