# Protograph LDPC Code Design for Asynchronous Random Access +
Footnote †: This work addresses the physical layer channel code design for an uncoordinated, frame- and slot-asynchronous random access protocol. Starting from the observation that collisions between two users yield very specific interference patterns, we define a surrogate channel model and propose different protograph low-density parity-check code designs. The proposed codes are both tested in a setup where the physical layer is abstracted, as well as on a more realistic channel model, where finite-length physical layer simulations of the entire asynchronous random access scheme, including decoding, are carried out. We find that the abstracted physical layer model overestimates the performance when short blocks are considered. Additionally, the optimized codes show gains in supported channel traffic, a measure of the number of terminals that can be concurrently accommodated on the channel, of around 17% at a packet loss rate of \(10^{-2}\) w.  
labelled as _modern random access_[6; 7; 8; 9; 10]. In all these schemes, nodes send proactively multiple copies of the same packet, while successive interference cancellation (SIC) enhances the receiver and helps in resolving contention. The pioneer of these schemes is contention resolution diversity slotted ALOHA (CRDSA) [6] (concurrently, the work in [11] also proposed a random access tree-resolution scheme that relies on the use of SIC at the receiver, which is able to largely outperform the classic standard and modified tree-algorithms [12; 13]) where the nodes are enforced to transmit two packet copies per transmission attempt [6]. The CRDSA protocol shows a great performance gain compared to SA. Tools borrowed from the theory of codes on graphs [14; 15] enable an asymptotic analysis of the performance [7], when the delay among replicas grows very large. The analysis suggests that a variable number of packet copies per user is beneficial. The probability mass function of these packet copies is called _user degree distribution_. Analysis of the finite length performance (when the delay among replicas becomes bounded) has shown that good user degree distributions discovered for the asymptotic setting perform well also in the bounded delay regime [16; 17]. A further extension was proposed in [8], where instead of simply _repeating_ the packets prior to transmission, a coded version is generated. Such a modification is able to achieve higher energy efficiency, compared to [7]. In [8], also an achievable throughput region for the collision channel was derived. Impressively, letting the maximum number of replicas and delay among them grow large is sufficient to achieve the limit of one packet per slot in the collision channel [18]. This remarkable result shows that such schemes are able to close the gap to coordinated and orthogonal medium access schemes. In [9], the authors showed that removing medium access (MAC) frame synchronization, while keeping slot synchronization, is beneficial to the performance, and they adopted a sliding-window receiver to cope with the frame-asynchronism among users. Along a similar line of research, in [10], the authors investigated the behaviour of a random access protocol with repetitions where the frame dimension is not set a priori, but it is dynamically adapted for maximizing the throughput. In [19], an information theoretic rate bound that considers the finite length nature of M2M communications and a more realistic channel model were derived. The paper shed light on the gap between practical schemes like CRDSA and the achievable bound, raising awareness to a larger community about the energy-efficiency problem. A new wave of research has been initiated, bringing several new approaches to the uncoordinated (and unsourced) multiple-access problem, e.g., [20; 21; 22; 23; 24]. The relaxation of slot-synchronicity reduces the complexity and especially power consumption of the transmitter devices.