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.

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