Analog Coding: Knots, Tori, Tube Packing, and Why The Digital Era May Not Last Forever
Digital error correction schemes can be viewed from the perspective of packing discrete points in Euclidean space and as such have firm roots in sphere packing, lattices, and groups. However, practical digital coding of continuous alphabet sources is now known to be suboptimal to analog coding from an information-theoretic perspective since finite block lengths must be employed. We discuss a new class of codes for analog error correction constructed as geodesics on flat tori that are within 6 dB of Shannon's optimal performance theoretically achievable using just a single symbol block length. We utilize the global circumradius function from knot theory and harmonic functions to design and optimize the code. We exploit the isometry of the flat torus with the hyperrectangle to derive simple closed-form decoders based on torus projections that come with 2 dB of the maximum likelihood decoder. We show simulation results and discuss the advantages of analog coding over digital coding.