# Feedback Approach for the Relay Channel with Noisy Feedback and Its Security Analysis
## 1 Introduction  
The relay channel (RC), which captures the essence of satellite communication [1, 2, 3], receives much attention in the literature. To explore the fundamental limit of the transmission over the RC, in recent years, the studies [4, 5, 6, 7, 8, 9, 10], respectively, have extended the _decode-and-forward_ (DF) strategy [11], the _compress-and-forward_ (CF) strategy [11], and the _amplify-and-forward_ (AF) strategy [12] for the single relay case to the RC. These extensions have provided corresponding lower bounds on the capacity of the RC according to these relay strategies. Nowadays, ultra-reliable and low-latency communication (URLLC) [13] serves as one of the key technologies in fifth-generation (5G) wireless communications since it supports many critical services in 5G that require a high level of reliability and low latency. Recently, the study of URLLC in the RC has received much attention, which mainly focuses on performance analysis [14, 15, 16] and emerging wireless access techniques [17, 18, 19], evaluating key metrics such as transmission rates, latency, and reliability to improve RC performance. However, there is a notable lack of attention to the design of efficient, low-complexity coding schemes for RC. This gap highlights the need for further exploration into coding strategies that can reduce computational demands while maintaining or improving system performance. Though channel feedback does not increase the memoryless channel capacity [20], it helps to construct an efficient coding scheme, such as the well-known Schalkwijk-Kailath (SK) scheme [21]. It has been shown that the SK scheme achieves a desired decoding error probability with extremely short coding blocklength [22], which indicates that it may be a good code for URLLC scenarios. Recently, ref. [23] extended the classical SK scheme [21] to the Gaussian relay channel (GRC) with noiseless feedback. In [23], the AF relay strategyand the classical SK coding scheme are combined, and to eliminate the impact of the relay noise on the performance of the original coding scheme, an auxiliary signal is generated, and the difference between the receiver's received signal and this auxiliary signal is used for the receiver's decoding. However, note that this SK-type scheme is based on noiseless feedback, which is unrealistic in practical communication scenarios. Then, it is natural to ask: can we extend the SK-type scheme in [23] to the noisy feedback case? On the other hand, due to the broadcast nature of wireless communication, in recent years, physical layer security (PLS), which started from [24], has been proven to be a useful tool to deal with the eavesdropping problem in wireless communication. In [25], it has already been shown that the classical SK [21] scheme achieves perfect weak secrecy by itself, where the perfect weak secrecy is known as one standard for the PLS. This self-secure property has been extensively studied in the literature; see [26; 27; 28]. However, we should note that this property holds for the noiseless feedback case, and whether it also holds for the noisy feedback case remains unknown. To answer the aforementioned questions, in this paper, we aim to extend the SK-type scheme to the GRC with noiseless feedback [23] to the wireless scenario and explore its security performance when an external eavesdropper is considered. First, we propose such an extended scheme for the additive white Gaussian noise (AWGN) RC in the presence of AWGN feedback, where the modulo-lattice function is introduced to mitigate the influence of noise within the feedback channel on the efficacy of the SK scheme. Furthermore, the utilization of multiple-input, single-output (MISO) technology in satellite relay communications offers significant advantages in enhancing the performance and reliability of the communication system [2; 3]. The above scenario can be modeled as using multiple transmitting antennas at the satellite and a single receiving antenna at the ground station and relay station.