# An LDPC-RS Concatenation and Decoding Scheme to Lower the Error Floor for FTN Signaling
## 1 Introduction
A typical bit error rate (BER) curve of channel coding resembles a waterfall, i.e., the BER drops sharply when the signal-to-noise ratio (SNR) increases. However, an error floor phenomenon can be found with many LDPC codes; when SNR increases, the BER does not decrease, or the curve flattens. According to [6], the error floor [7] is caused by sub-optimal iterative decoding [8], near-code words [9], and absorbing and trapping sets [10]. In some situations, the error floor can not be eliminated, and there are some methods to lower the error floor, for example, designing a large-girth parity matrix to optimize the code structure; using concatenation code, such as RS-LDPC [11] or DVB BCH-LDPC code [12] that utilizes RS or BCH as the outer code and the LDPC as the inner code; or manually introducing low-magnitude noise to message-passing decoders to lower the error floor. For some conditions, such noise can actually improve decoding performance [13]. This improvement, based on the effect of noise, is called stochastic resonance. We can refer to the introduction of such noise as perturbation. Perturbation decoders have exceptional performance and very low resource costs [14] and may prove superior to the existing techniques [15] in computer memory and data storage applications [16]. Details about stochastic resonance can be found in [17]. Note that the error floor may occur in a particular code and may not occur in another, and it is hard to predict whether it will occur. For example, the 5G LDPC (7680, 3840) code used in this study has no error floor in orthogonal (Nyquist) systems. However, when utilizing FTN signaling, we found an error floor in a rate-1/2 LDPC code masked from the 5G base graph \"BG2\". This is probably caused by the decoder's non-linearity, too few turbo iterations, improper loop gain or code rate, or a weakness in the masked base matrix structure. Since this phenomenon exists and is unpredictable, we require a method to lower the error floor that may occur in the demand region of the BER for practical situations. In this work, we utilized two of the above methods: LDPC-RS concatenated code and a perturbation BP decoder to lower the error floor. Reed-Solomon (RS) code was invented by [PERSON] and [PERSON] [18]. RS code has undergone decades of development and has been used in wireless, satellite, and optical communications due to its maximum distance separable (MDS) feature. A hard decision decoder is often utilized for high-speed or concatenated code due to its simple structure and high throughput. In this study, we used soft decision LDPC code due to its excellent error correction ability concatenated with hard decision RS code because of its high decoding speed and definite error correction ability. Our contribution in this paper can be summarized in two points: (1) To combat the error floor, we propose one possible solution, the use of the LDPC-RS concatenation and shortening scheme. We introduce perturbation into the BP decoder to enhance its performance. (2) We propose a 4-parallel encoder architecture for RS component code and a concise algorithm to calculate its constant multiplier coefficients, leveraging a traditional serial encoder, which can also be used for calculating other parallelisms, code rates, and lengths. The simulation results show that the proposed concatenation and shortening scheme can lower the error floor of the original single LDPC code, and with perturbation in the BP decoder, the error floor can be eliminated below \(10^{-7}\) for different LDPC maximum decoding iterations. The proposed scheme has an error correction ability for BCJR detection with few states and few iterations of turbo equalization FTN signaling and successfully lowers the error floor. The outline of this paper is as follows: Section 2 reviews preliminaries of FTN signaling with the OBM, BCJR detection, turbo equalization, LDPC, and RS codes, as well as our simulations with 5G LDPC code and an error floor.