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

Consider the real additive white Gaussian noise sequence channel

Y=X+Z,Y=X+Z,

where the noise coordinates are independent N(0,N0/2)N(0,N_0/2). An (E,M,ϵ)(E,M,\epsilon) code has MM equiprobable messages, a deterministic encoder with codewords c1,…,cM∈ℓ2c_1,\ldots,c_M\in\ell_2 satisfying

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

no feedback, and an arbitrary decoder with average block error probability at most ϵ\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,ϵ)E^*(k,\epsilon) for every fixed kk and ϵ\epsilon.

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

Progress summary

Solved

A 2026 preprint claims the conjecture is settled: the equally spaced simplex code is optimal, but the proof has not yet been independently checked.

The problem asks whether the full-energy regular (M−1)(M-1)-simplex minimizes the energy needed to transmit MM equiprobable messages over the no-feedback AWGN channel. Polyanskiy and Wu presented this as the fixed-message conjecture in Remark 21.2 of their book.

Known results

  • Pulse-position modulation, after centering and rescaling, yields a regular simplex; its non-asymptotic optimality was previously conjectured, not proved.

August 2026 claimed proof

A preprint titled “Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture” claims a complete all-dimensional proof. It asserts that the radius-a0Ea0\sqrt{E} regular simplex maximizes correct-decoding probability and therefore gives the exact fixed-(M,ϵ)(M,\epsilon) minimum energy, including zero energy for ϵ≥1−1/M\epsilon\ge 1-1/M and infinite energy for ϵ=0\epsilon=0. The preprint does not establish uniqueness, and no independent verification or reported objection was found.

Current status (as of August 2026): A preprint claims the simplex is exactly optimal for the stated deterministic, equiprobable, maximal-energy AWGN problem, but this remains an unverified claimed resolution; uniqueness is open.

Sources
Sources & referencesView supporting material

References

Yury Polyanskiy, H. Vincent Poor, Sergio Verdú, Minimum energy to send kk bits through the Gaussian channel with and without feedback. Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Remark 21.2.

awgncoding-theoryfinite-blocklengthgaussian-noiseinformation-theorysimplex-conjecture
Posted by Abhijeet Mulgund · edited

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