# Design of Chaotic Interleaver Based on Duffing Map for Turbo Code
## 1 Introduction  
Convolutional codes concatenated in parallel are used for error detection and correction, as well as denominated turbo codes created by [PERSON], [PERSON], and [PERSON] in 1993 [1]. In terms of bit error rate, the performance of convolutional codes concatenated in parallel is close to Shannon's limit. Shannon's channel capacity is a superior quota that establishes the maximum quantity of digital data that can be transmitted without error over a communication link that has a specific bandwidth and is subject to noise interference. This outstanding characteristic has allowed for the incorporation of turbo codes in communication standards such as 3G Long Term Evolution (LTE) [2], LTE/LTE-A for 4G mobile communication systems [3], and 5G communications [4]. For the 5G communication standard, its application considers diverse scenarios and would not be restricted to the coverage of large areas but would also be employed in machine-to-machine, vehicle-to-vehicle, indoor, and outdoor communications, among others [5]. Undoubtedly, the 5G network will present a significant change in the industry, businesses, and daily life in general. The coverage, data performance, latency, and energy efficiency will improve compared with the current fifth-generation system [6]. However, the diversity of application scenarios indicates that there will be multiple 5G technologies, which would not be represented by one dominant technology as they would coexist with some previous technologies from 1G to 4G systems. Currently, applications such as machine communication, real-time communication, the Internet of Things, wireless sensor networks, smart metering networks, satellite uplink, messenger services, and remote command links [7; 8; 9; 10] require short and mediumdata lengths, i.e., in the order of 50 to 1000 bits, in the communication [11]. Because of some emerging applications, interest in short and medium blockchain turbo codes has increased [12; 13; 14]. The applications above are characterized by having specific requirements such as low-latency and high-reliability transmitters. The use of short packages is considered a key element in complying with these requirements [15]. Other codes, such as the low-density parity check (LDPC) presented in [16] and the polar code, have similar error correction capacities [17]. Nevertheless, the computing complexity of both LPDC and polar decoders is lower for most codification rates, as opposed to the high complexity required in turbo codes [18]. Turbo, LDPC, and polar codes admit different block lengths L and coding rates R, which are given by their structures. In experimental coder tests for short lengths (64 and 128 bits), the turbo code performs better than the LDPC code [11]. The design of long channel codes for iterative decoding is well established and based on a set of performance metrics, such as the iterative decoding threshold [19] or the ensemble typical minimum distance [20]. For larger blocks, modern channel codes such as turbo codes [9] have an excellent performance in several communication channels. However, nowadays, performance degradation associated with short block length turbo codes has restricted their use to applications that employ communications with short block-length codewords [10]. Most classic tools developed for turbo code design are not applicable to the design of short blocks that can be decoded iteratively because of their extremely low performance [11]. In connection, if the performance of turbo codes is correlated with the code's distance spectrum, an \"error floor\" is obtained, which is a consequence of the relatively small free distance of the code [21]. The design of an algorithm for decoding short blocks with optimal performance in terms of error correction and computational complexity continues to be an unsolved problem [10].