Recent research in ultra-reliable and low latency communications (URLLC) for future wireless systems has spurred interest in short block-length codes. In this context, we analyze arbitrary harmonic bandwidth (BW) expansions for a class of high-dimension constant curvature curve codes for analog error correction of independent continuous-alphabet uniform sources. In particular, we employ the circumradius function from knot theory to prescribe insulating tubes about the centerline of constant curvature curves. We then use tube packing density within a hypersphere to optimize the curve parameters. The resulting constant curvature curve tube (C3T) codes possess the smallest possible latency, i.e., block-length is unity under BW expansion mapping. Further, the codes perform within $5$ dB signal-to-distortion ratio of the optimal performance theoretically achievable at a signal-to-noise ratio (SNR) $< -5$ dB for BW expansion factor $n \leq 10$. Furthermore, we propose a neural-network-based method to decode C3T codes. We show that, at low SNR, the neural-network-based C3T decoder outperforms the maximum likelihood and minimum mean-squared error decoders for all $n$. The best possible digital codes require two to three orders of magnitude higher latency compared to C3T codes, thereby demonstrating the latter's utility for URLLC.

翻译：未来无线系统中超可靠低延迟通信（URLLC）的最新研究激发了人们对短分组码的兴趣。在此背景下，我们针对一类高维恒定曲率曲线码，分析了用于独立连续字母表均匀源模拟纠错的任意谐波带宽（BW）扩展。具体而言，我们利用纽结理论中的外接半径函数，在恒定曲率曲线的中心线周围规定绝缘管状结构，并采用超球体内的管填充密度来优化曲线参数。由此产生的恒定曲率曲线管（C3T）码具有最小可能的延迟，即在带宽扩展映射下分组长度等于1。此外，当带宽扩展因子$n \leq 10$时，该码在信噪比（SNR）$< -5$ dB条件下的信号失真比与理论最优性能相差$5$ dB以内。我们还提出了一种基于神经网络的C3T码解码方法。结果表明，在低信噪比下，对于所有$n$，基于神经网络的C3T解码器均优于最大似然解码器和最小均方误差解码器。与C3T码相比，最佳数字码需要高出两到三个数量级的延迟，从而证明了C3T码在URLLC中的实用性。