Trifina Lucian, Tarniceriu Daniela, Ryu Jonghoon, Rotopanescu Ana-Mirela
Department of Telecommunications and Information Technologies, "Gheorghe Asachi" Technical University, 700506 Iasi, Romania.
Samsung Electronics, Inc., Suwon 16677, Korea.
Entropy (Basel). 2020 Jan 8;22(1):78. doi: 10.3390/e22010078.
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.
在本文中，我们针对使用特定交织器长度的四阶置换多项式（4 - PP）交织器以及标称编码率为1/3的经典Turbo码，得到了最小距离的上界。该Turbo码具有两个生成矩阵为G = [1, 15 / 13]的递归系统卷积分量码。交织器长度为16Ψ或48Ψ的形式，其中Ψ是大于3的不同质数的乘积。当对于某个质数(p_i\mid\Psi)满足条件3 ∤ ((p_i - 1))时，会应用一些系数限制。对于不同类别的4 - PP系数，得到了两个上界。对于一个4 - PP (f_4x^4 + f_3x^3 + f_2x^2 + f_1x\ (\text{mod}\ 16k_L\Psi))，(k_L\in{1, 3})，当等效4 - 置换多项式（PPs）的系数(f_3)满足(f_3\in{0, 4\Psi})，或者当(f_3\in{2\Psi, 6\Psi})且(f_2\in{(4k_L - 1)\cdot\Psi, (8k_L - 1)\cdot\Psi})，(k_L\in{1, 3})，而其他系数为任意值时，得到上界28。当等效4 - PPs的系数(f_3)满足(f_3\in{2\Psi, 6\Psi})且(f_2\in{(2k_L - 1)\cdot\Psi, (6k_L - 1)\cdot\Psi})，(k_L\in{1, 3})，其他系数为任意值时，得到上界36。因此，由于系数(f_4)、(f_3)和(f_2)所需的范围较小，该结果极大地促进了寻找上述长度的良好4 - PP交织器的任务。通过非线性度还证明了，对于所考虑的交织器长度，具有最优最小距离的三次PPs和二次PPs相比于具有最优最小距离的4 - PPs会导致更好的误码率性能。
