# Design of Chaotic Interleaver Based on Duffing Map for Turbo Code
## 1 Introduction
The error floor that may appear in these cases, which varies according to the distribution of code words with low weight, can be reduced through an increase in the free distance or a reduction in the number of code words [21]. Commonly, this can be achieved after an improvement in the design of the interleaver [22]. However, this topic is still a new field of research at both the theory and technology levels [11], despite the significant advances in the design and analysis codes for correcting errors in short blocks made by the academy and the industry [23]. A review of the main studies on interleavers for turbo codes with short and medium block lengths is presented below. In [22], the design of parallel turbo codes with short block lengths in the order of 64 to 128 bits is dealt with. This work methodically describes how to build turbo codes with a large minimum distance for short block lengths and arbitrary rates. The interleaver parameters were adequately selected to maximize the minimum distance of the turbo code, such that the error floor is considerably reduced, and better performance is obtained in the waterfall region. However, the decoding algorithms designed are too complex for many applications. Quadratic permutation polynomial (QPP) interleavers have been widely researched since 2005. In [23], a study on permutation polynomial (PP) interleavers, which are deterministic interleavers for turbo codes, is conducted, and a degree 3, 4, and 5 PP is designed, which has optimal minimum distances and lowest corresponding codeword multiplicities for turbo codes with recursive convolutional component codes from LTE standard. The short and medium lengths used range from 40 to 512 bits. In [24], a new method for designing efficient puncture-constrained interleavers for turbo codes is created. The interleaver function is defined through a layered design process that considers several criteria, such as minimum span, correlation girth, and puncturing constraints. This new design method enables the resulting turbo codes to profit from a joint optimization of the puncturing pattern, so the interleaver achieves an improved error rate performance. The tests consider data blocks with 1504 bits. Their bit error rate (BER) is assessed on the basis of \(E_{b}/N_{0}\) (bit signal-to-noise ratio (SNR)), and the (symbol SNR) relation as a function of data between 100 and 1000 bits. Additionally, significant improvements are achieved in the waterfall and error floor regions thanks to the incorporation of puncturing constraints into the interleaver design. In [25], a modified interleaver design is shown, which takes advantage of the coder structure and improves the performance of the turbo code with a short block length. The applied criterion leads to the introduction of the subvector interleaver and of the S-random interleaver constrained by a subvector. The tests conducted use the 140- and 224-bit lengths and implement the corresponding turbo code in the 3 GPP LTE encoder with the proposed interleaver in the GNU radio platform. In [11], a comparison of the codes for short block lengths is presented. In addition, some of the most promising code buildings, which are aimed at the short block regimes, are introduced and compared with the performance limits for finite length as well as with the coding schemes for classic errors. The codes analyzed are turbo, LDPC, and polar codes, among others, which have block lengths of 512 and 256 bits with a one-half rate. From this analysis, it is concluded that the design of block codes for short data blocks is an unsolved research problem. In [26], the turbo codes of the LTE standard are considered. These codes have short and medium interleaver lengths with 16\(\Psi\) or 48\(\Psi\) forms, where \(\Psi\) is a product between different prime numbers bigger than 3. The analyzed interleavers are Cubic Permutation Polynomials (CPP), for which the minimum distance is constrained by the 38, 36, and 28 values for three different classes of polynomial coefficients.