Constant Curvature Curve Tube Codes for Low-Latency Analog Error Correction

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 introduce a new class of high-dimension constant curvature curves 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 -- block-length is unity under bandwidth expansion mapping. Further, the codes provide within $5$ dB of Shannon's optimal performance theoretically achievable at the lower range of signal-to-noise ratios and BW expansion factors. We exploit the fact that the C3T encoder locus is a geodesic on a flat torus in even dimensions and a generalized helix in odd dimensions to obtain useful code properties and provide noise-reducing projections at the decoder stage. We validate the performance of these codes using fully connected multi-layer perceptrons that approximate maximum likelihood decoders. For the case of independent and identically distributed uniform sources, we show that analog error correction is advantageous over digital coding in terms of required block-lengths needed to match {signal-to-noise ratio, source-to-distortion ratio} tuples. 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.