## 1 Introduction
### Related Work and Motivation  
The design of low-resolution quantized precoders for large-scale MIMO systems, particularly in downlink scenarios, has garnered significant attention in recent research [14; 15]. Initially, the focus was on block-level linear precoding techniques, but the field has since evolved towards symbol-level nonlinear precoding methods. Linear precoders such as zero-forcing (ZF), maximal-ratio transmission (MRT), and Wiener filter (WF) remain popular due to their straightforward implementation and low computational demands [14]. These methods perform competitively when applied to systems with large antenna arrays. However, the integration of low-resolution digital-to-analog converters (DACs) at the base station (BS) can constrain the signal's degrees of freedom. For example, the authors in [15] proposed an optimization strategy based on minimum mean square error (MMSE), which demonstrated improved bit error rate (BER) performance over traditional ZF precoders. Nevertheless, despite their efficiency, linear quantized precoders may struggle to maintain consistent and reliable transmission quality, particularly when dealing with very coarse one-bit quantization levels. In the realm of nonlinear precoding, the authors in [16] proposed a novel method for designing transmit signals in large-scale MIMO systems with one-bit DACs by minimizing the MMSE between the generated spatial spectrum and the target spectrum. This approach formulates the problem as a fourth-order non-convex optimization with discrete constraints, which is tackled using an alternating minimization technique that simplifies the objective to a quadratic form through auxiliary variables. Similarly, the authors in [8] utilized the squared-infinity norm Douglas-Rachford splitting (SQUID) method to develop an algorithm for one-bit DACs, albeit with significant computational overhead. Other notable works include [17], which employed convex relaxation for nonlinear precoding with one-bit DACs, and [18], which introduced an efficient nonlinear precoder based on the alternating direction method of multipliers (ADMM), reducing complexity via matrix decomposition. Building on these, the authors in [19] presented an \(\ell_{0}\)-ADMM algorithm to further decrease power consumption by adding degrees of freedom. Additionally, the authors in [20] proposed lattice reduction-aided continuous interference cancellation to enhance the searchability of the Babai nearest plane without requiring parameter adjustments. To improve error rate performance, nonlinear one-bit precoding designs were introduced in [21; 22], where data symbols were directly mapped to one-bit transmission signals using symbol-level operations. Recent advancements have demonstrated that frame-level precoding with low-resolution DACs (typically 2-4 bits) can achieve performance comparable to high-resolution systems [23]. This has sparked interest in symbol-level precoding techniques utilizing coarse-resolution DACs. Researchers have explored various optimization criteria for this approach, including MMSE [24; 25; 26; 27], minimum symbol error rate (SER) [28], and maximum signal-to-noise ratio (SNR) [29]. Notably, the authors in [24] introduced an MMSE-based quantized symbol-level precoding method employing cyclic coordinate descent. While large-scale multi-user MIMO (MU-MIMO) systems offer enhanced communication efficiency and spectral utilization, integrating radar sensing functionalities within these systems, especially under low-resolution DAC constraints, remains challenging. Despite these advances, the integration of radar sensing and communication in large-scale MIMO DFRC systems introduces new challenges. High-resolution DACs, although effective, result in substantial hardware costs and power consumption--factors that limit system scalability. Low-resolution DACs, particularly one-bit DACs, offer promising cost and energy benefits but severely restrict the degrees of freedom in waveform design. This quantization leads to performance degradation if not carefully managed.