# A New Coding Paradigm for the Primitive Relay Channel +
## 1 Introduction  
The relay channel, introduced by [PERSON] in [1], represents the simplest network model with a single source and a single destination. The source wants to communicate with the destination, and the relay helps the communication. More specifically, let \(X_{\mathrm{s}}\) be the signal sent by the source to the relay and to the destination, \(Y_{\mathrm{ss}}\) the signal received by the relay, \(X_{\mathrm{s}}\) the signal sent by the relay to the destination, and \(Y_{\mathrm{o}}\) the signal received by the destination which comes from the source and from the relay. Note that the relay channel has a broadcast component going from the source to the relay and to the destination, and a multiple access component going from the source and from the relay to the destination. The model is schematized in Figure 1.  
[PERSON] and [PERSON] provided a general upper bound (the cut-set bound) and two lower bounds (decode-and-forward and compress-and-forward) in [2]. Since that seminal work, several lower bounds have been derived, i.e., amplify-and-forward, compute-and-forward, noisy network coding, quantize-map-and-forward, hybrid coding, see [3; 4; 5; 6; 7]. The cut-set bound is tight in most of the settings where capacity is known [2; 8; 9; 10]. However, the cut-set bound was shown not be tight in some special cases [11; 12], and novel upper bounds tighter than cut-set were recently presented in [13; 14; 15; 16]. For a review on the relay channel, see also ([17] Chapter 16) and ([18] Chapter 9). Polar codes, introduced by [PERSON] in [19], have been employed to devise practical schemes for the relay channel. In particular, for the case of the degraded relay channel where \(X_{\mathrm{s}}\to(X_{\mathrm{k}},Y_{\mathrm{sR}})\to Y_{\mathrm{0}}\) forms a Markov chain, polar coding techniques for decode-and-forward are presented in [20; 21; 22; 23]. Furthermore, for the case of the relay channel with orthogonal receiver components, a polar coding scheme for compress-and-forward is proposed in [22]. For general relay channels, polar coding techniques for decode-and-forward and compress-and-forward are described in [24]. We will adopt these schemes as primitives in our approach. Soft decode-and-forward relaying strategies which employ low-density parity-check (LDPC) codes are considered in [25]. In this work, we consider the relay channel with orthogonal receiver components, which is also known as the primitive relay channel. The difference with respect to the general relay channel consists in the fact that the destination receives two separate signals: \(Y_{\mathrm{sD}}\) from the source and \(Y_{\mathrm{sD}}\) from the relay. Basically, the multiple access component going from the source and from the relay to the destination is substituted by two parallel channels. Furthermore, we assume that the relay can listen and transmit simultaneously, namely, it is full-duplex. The model is schematized in Figure 2. Note that the relay communicates with the destination via a direct link. Thus, the relay can communicate reliably to the destination at a rate arbitrarily close to capacity by using a capacity achieving code (e.g., a random code or a polar code). Consequently, we can just assume that the relay and the destination are connected via a noiseless link of given capacity. Even in this simplified setting, the capacity of the primitive relay channel is unknown in general. A review on coding scheme for the primitive relay channel is contained in [26]. The main contribution of this paper is a novel coding scheme that combines compress-and-forward with decode-and-forward and improves upon both of them. The idea is to consider pairs of blocks and use a chaining construction: in the first block, we perform a variation of compress-and-forward where the relay sends only a part of the compressed signal to the destination; in the second block, we perform decode-and-forward and the relay sends to the destination the new information bits together with the remaining part of the compressed signal coming from the previous block.