Exact finite-energy fundamental limit for deterministic no-feedback AWGN coding

Question

Consider the real additive white Gaussian noise sequence channel

Y=X+Z,

where the noise coordinates are independent $N(0,N_0/2)$ . An $(E,M,\epsilon)$ code has $M$ equiprobable messages, a deterministic encoder with codewords $c_1,\ldots,c_M\in\ell_2$ satisfying

\|c_i\|^2\le E \qquad(1\le i\le M),

no feedback, and an arbitrary decoder with average block error probability at most $\epsilon$ . There is no restriction on the number of channel uses or nonzero coordinates.

Polyanskiy and Wu observed in Remark 21.2 of Information Theory: From Coding to Learning that pulse-position modulation can be improved by subtracting its common center of gravity and rescaling, producing a regular simplex code. They conjectured that this simplex code is the actual optimal code achieving $E^*(k,\epsilon)$ for every fixed $k$ and $\epsilon$ .

Equivalently, for $M=2^k$ , does the full-energy regular $(M-1)$ -simplex minimize the required total energy among all deterministic, equiprobable, no-feedback AWGN codes satisfying the maximal per-codeword energy constraint?

Exact finite-energy fundamental limit for deterministic no-feedback AWGN coding

References

Progress summary

Solutions 1