# Protograph LDPC Code Design for Asynchronous Random Access +
Avoiding the need to keep synchronization to a common clock enables longer _sleep_ times for the nodes. This is of uttermost importance in many M2M scenarios, where devices are powered via batteries that cannot (or may not) be replaced for their entire lifetime. Some recent solutions derived from the _asynchronous_ ALOHA protocol have been shown to be competitive in comparison with their slot-synchronous counterpart [25; 26; 27; 28]. These solutions also adopt the transmission of multiple copies of the packets and SIC at the receiver. When the receiver entangles combining techniques, e.g., maximal-ratio combining, with SIC, slot-synchronous protocols may be outperformed [27] (the performance heavily depends on the specific configuration, i.e., number of replicas, physical layer forward error correction, channel conditions, etc.). In this work, we consider an asynchronous random access scheme. We assume that every packet (codeword) is subject to additive white Gaussian noise (AWGN) and possibly to interference. Due to the asynchronous nature of the scheme, the interference may affect different portions of the packet with different power, depending on the number of colliding users. In order to allow reliable transmission on the asynchronous random access channel, a suitable error protection scheme needs to be used. Works in the literature usually assume capacity achieving random code ensembles and apply a threshold-based model for decoding [25]: the average signal-to-noise plus interference ratio over a packet is compared to the Shannon limit, i.e., the worst channel parameter for which error-free transmission is possible, to decide whether decoding is possible or not. In [29], the authors replaced the Shannon limit by iterative decoding thresholds of some off-the-shelf low-density parity-check (LDPC) code ensembles. In an earlier conference paper [30], we considered a decoding region (which can be seen as a multi-dimensional threshold-based model) to perform dedicated protograph-based LDPC code design for the asynchronous random access channel and to estimate the packet loss rate (PLR) of the random access protocol. However, it remains an open question how accurately such threshold-based models can predict the PLR. The code design of [30] using a multi-dimensional threshold-based model can be seen as a type of code design for unequal error protection. Unequal error protection using error correcting codes has long been studied in the literature for different channels. Early works date back to the 1960s. For instance, the work in [31] addressed the problem of achieving different bit error probabilities in different parts of the decoded codeword. More recently, both turbo codes and LDPC codes have been designed for channels that introduce different reliabilities to different parts of the codeword [32; 33; 34] to counteract the effect of block fading: a block is characterized by a constant fading coefficient, while block-by-block, the fading coefficient is independently drawn from a predefined distribution. In the following, we focus on the asynchronous random access channel for which we provide tailored protograph LDPC code designs. Here, we make use of a surrogate channel model to simplify the code design. We show that the resulting codes evidently boost the MAC performance, even in comparison with the most competitive code designs for standard AWGN channels. As an extension of [30], we do not restrict to finding LDPC code ensembles with favourable iterative decoding thresholds only, but also design finite-length codes. These codes are used to simulate the physical layer of the random access protocol, giving a more realistic estimate of the PLR. The key contributions of the paper can be summarized as follows:  
* In Section 3, we present a surrogate channel model, exploited in the code design phase, which assumes constant interference power over a fraction \(\alpha\) of the codeword.