## 1 Introduction  
Modern wireless vehicular networks are highly dynamic, and the connectivity of such networks is heavily dependent on key techniques, like relaying. Relaying is an important strategy in wireless vehicular networking, and therefore, it has received much attention in recent years. Physical layer network coding (PNC) is a new paradigm of relaying [1] in which intermediate nodes in a wireless network, instead of decoding individual symbols from the received signals, decode a linear combination of symbols and send them to the destination node. Upon obtaining sufficient combinations, the destination node can decode the original signals. In [2], [PERSON] and [PERSON] proposed a novel strategy of generalized relaying, called compute and forward (CF), which enables the relays in any Gaussian wireless network to decode linear equations of the transmitted symbols with integer coefficients, using the noisy linear combinations provided by the channel. The linear equations are transmitted to the destination, and upon receiving sufficient linear equations, the destination can decode the desired symbols. The key point in this strategy is the use of nested lattice codes for encoding the original messages. Nested lattice codes satisfy the property that a linear combination of codewords gives another codeword. Further, information theoretical tools are used in [2] to obtain the achievable rate regions. An algebraic approach to implement CF has been introduced in [3]. In this work, the authors have related the approach introduced by [PERSON] and [PERSON] to the fundamental theorem of finitely generated modules over the principal ideal domain (PID). Consequently, the isomorphism between the message space and the physical signal space (equivalently between the link layer and physical layer) is identified using module theory. Furthermore, the authors identify the lattices and lattice partitions, which can be utilized to implement CF in finite dimensions. The authors show that the union bound estimate for the probability of decoding a linear equation at the relay node is limited by the gap between the channel gain and the integer approximation. In [4], [PERSON] and [PERSON] have analyzed the asymptotic behavior of CF and pointed to a fundamental limitation. They have shown that the degrees of freedom (DoF) achievable by the lattice-based implementation of CF is at most two, when the channel gains are irrational. This is due to the gap between the natural channel gains and the integer coefficient of the linear combination in coherence to the observation in [3]. Further, it is proved that this limitation is not inherent to the fundamental concept of CF, but is due to the lattice-based implementation of CF. In addition, it has also been shown in [4] that the maximum DoF can be achieved by CF when channel state information (CSI) is available at the transmitter. Another recent work in [5] studies the practical aspects of CF. Specifically, the decoding techniques are studied, and it is shown that the additional noise created due to the non-integer channel coefficients makes the effective noise non-Gaussian and increases the complexity of Maximum likelihood decoding in CF. The study of CF has attracted the attention of the research community, and a wide body of work is available in the literature. Interested readers can refer to an extensive survey of the recent developments in CF in [6]. Different implementations of CF have also been proposed. The selection of integer coefficients of linear combination at the relay has been studied in [7], where the authors provide ascheme to choose sub-optimal integer coefficients without using CSI at the relay. Further, in [8, 9], precoding-postcoding techniques have been studied in order to minimize the effects of the non-integer channel penalty in some scenarios, particularly, using the spatial orientation of signals. Motivated by these results, in this paper, we aim to remove the decoding limitations of the previous proposals, focusing on the physical layer.