Analysis of error correction performance for error correcting codes is very important when using such codes in digital communication systems. At medium-to-high signal-to-noise ratios, the distance spectrum of the error correcting code represents a good indicator for the error correction performance of the code. It is desired that the minimum distance of the code is as large as possible and that the corresponding multiplicity (i.e. the number of codewords having the weight equal to the minimum distance) is as small as possible. If we know an upper bound of the minimum distance of the code, then we have a good indication about the capabilities and the limitations of the code. One of the classes of the error correcting codes with the best performance is that of turbo codes. For such codes, establishing upper bounds on the minimum distance is challenging because it depends on the interleaver component of the turbo code. In this paper we consider turbo codes with component convolutional codes as in the Long Term Evolution standard. The interleaver lengths are of the form \(16 \varPsi \) or \(48 \varPsi \), with \( \varPsi \) a product of different prime numbers greater than three. The first achievement in the paper is that for these interleaver lengths, we show that cubic permutation polynomials (CPP), with some constraints on the coefficients, when \(3 \not \mid (p_i-1)\) for a prime \(p_i > 3\), always have a true inverse CPP. The most accurate upper bounds on the minimum distance for turbo codes are achieved by identifying bit information sequences leading to a certain weight of the corresponding turbo-codeword. In this paper we have indentified such bit information sequences by means of the full range dual impulse method to estimate the weight of the turbo-codewords. For the previously mentioned turbo codes and CPP interleavers, we show that the minimum distance is upper bounded by the values of 38, 36, and 28, for three different classes of coefficients. Previously, it was shown that for the same interleaver lengths and for quadratic PP (QPP) interleavers, the upper bound of the minimum distance is equal to 38. Several examples show that \(d_{min}\)-optimal CPP interleavers are better than \(d_{min}\)-optimal QPP interleavers because the multiplicities corresponding to the minimum distances for CPPs are about a half of those for QPPs. A theoretical explanation in terms of nonlinearity degrees for this result is given for all considered interleaver lengths and for the class of CPPs for which the upper bound is equal to 38.

在数字通信系统中使用纠错码时，对纠错性能进行分析非常重要。在中等到高的信噪比下，纠错码的距离谱代表了该码的纠错性能的良好指标。期望代码的最小距离尽可能大，并且相应的多重性（即，权重等于最小距离的代码字的数量）尽可能小。如果我们知道代码最小距离的上限，则可以很好地了解代码的功能和局限性。具有最佳性能的一类纠错码是turbo码。对于此类代码，在最小距离上确定上限是有挑战性的，因为它取决于turbo码的交织器组件。在本文中，我们将采用长期演进标准中的带有分量卷积码的Turbo码。交织器长度的形式为\（16 \ varPsi \）或\（48 \ varPsi \），其中\（\ varPsi \）的不同质数大于3的乘积。在纸张上的第一成就是，对于这些交错器的长度，我们表明，立方置换多项式（CPP），对所述系数的一些限制，当\（3 \不\中期（P_I-1）\）为素数\（ p_i> 3 \），总是有一个真正的逆CPP。通过识别导致相应turbo码字的一定权重的比特信息序列，可以实现turbo码最小距离的最精确上限。在本文中，我们已通过全范围双脉冲方法识别了此类比特信息序列，以估计Turbo码字的权重。对于前面提到的turbo码和CPP交织器，我们表明，对于三种不同类别的系数，最小距离由值38、36和28上限。以前，已经证明，对于相同的交织器长度和二次PP（QPP）交织器，最小距离的上限等于38。几个示例表明\（d_ {min} \）-最佳CPP交织器更好比\（d_ {min} \）-最佳QPP交织，因为与CPP的最小距离对应的多重性约为QPP的一半。对于所有考虑的交织器长度和上限等于38的CPP类，给出了针对该结果的非线性度的理论解释。