Polynomial Freiman-Ruzsa, Reed-Muller codes and Shannon capacity

In 1948, Shannon used a probabilistic argument to show the existence of codes achieving a maximal rate defined by the channel capacity. In 1954, Muller and Reed introduced a simple deterministic code construction based on polynomial evaluations, which was conjectured and eventually proven to achieve capacity. Meanwhile, polarization theory emerged as an analytic framework to prove capacity results for a variation of RM codes - the polar codes. Polarization theory further gave a powerful framework for various other code constructions, but it remained unfulfilled for RM codes. In this paper, we settle the establishment of a polarization theory for RM codes, which implies in particular that RM codes have a vanishing local error below capacity. Our proof puts forward a striking connection with the recent proof of the Polynomial Freiman-Ruzsa conjecture [40] and an entropy extraction approach related to [2]. It further puts forward a small orbit localization lemma of potential broader applicability in combinatorial number theory. Finally, a new additive combinatorics conjecture is put forward, with potentially broader applications to coding theory.

💡 Research Summary

The paper establishes a polarization theory for Reed–Muller (RM) codes that leads to a new proof of their weak capacity-achieving property on the binary symmetric channel (BSC). Historically, Shannon showed that random codes can achieve the channel capacity 1 – H(δ) where δ is the bit‑flip probability, but constructing explicit deterministic codes with this property remained a major challenge. While polar codes provided a clean analytic framework (polarization) to prove capacity for a class of codes related to RM, a full polarization result for RM codes themselves was missing.

The authors bridge this gap by connecting three seemingly disparate ingredients: (i) the recent proof of the Polynomial Freiman‑Ruzsa (PFR) theorem, which asserts that if the entropy increase under addition of two independent binary vectors is small then the vectors are close to being uniformly distributed on a large subspace; (ii) an entropy‑extraction viewpoint that tracks the conditional entropies of successive layers of the RM code (the coefficients of monomials of degree ≤ r); and (iii) a new “small orbit localization lemma” that guarantees the existence of a subspace invariant under a family of linear transformations and that is, on average, close to a randomly drawn subspace.

The main technical result (Theorem 3.1) consists of three parts. First, a layer‑polarization inequality shows that the conditional entropy aₘ,ᵣ = H(U_{≤r} | Y, U_{>r}) satisfies a recursive bound involving the entropy gaps fₘ,ᵣ = aₘ,ᵣ – aₘ,ᵣ₋₁. This inequality is derived by applying the PFR theorem to the conditional distributions of the RM coefficients, thereby guaranteeing that the entropy gaps decay at a controlled rate as the degree r increases. Second, assuming the code rate stays below capacity (i.e., lim supₘ (∑_{i≤rₘ} C(m,i))/2ᵐ = (1 – ε)(1 – H(δ)) for some ε > 0), the authors prove that aₘ,ᵣₘ ≤ 2ᵐ/2 – c √m for a constant c depending on ε. This gives a quantitative bound on the residual uncertainty of the low‑degree coefficients after observing the noisy codeword and the high‑degree coefficients (which are set to zero in RM). Finally, they translate this entropy bound into an error‑probability bound for the maximum‑likelihood decoder: the bit‑error probability decays as 2^{–Ω(√m)}. Consequently, the RM family {RM(m, rₘ)} achieves Shannon capacity in the weak sense (vanishing bit error) for any rate below capacity.

A central combinatorial tool is Lemma 3.2, the small orbit localization lemma. It states that for any distribution W over subspaces of F₂ⁿ that is invariant under a family 𝒯 of linear transformations, there exists a subspace G* fixed by all T ∈ 𝒯 such that the expected distance between a random subspace drawn from W and G* is at most half the expected distance between two independent draws from W. In the context of Boolean polynomial spaces, the only subspaces invariant under all affine linear maps are the trivial subspace and the whole space, which yields strong lower bounds on the distance to G*. This lemma enables the authors to replace arbitrary random subspaces by highly structured invariant ones, simplifying the entropy analysis.

Beyond the main result, the paper proposes a new additive‑combinatorics conjecture that generalizes the orbit‑localization phenomenon to broader families of symmetric codes. The conjecture suggests that similar invariant‑subspace arguments could yield polarization for other algebraic codes (e.g., BCH, Reed‑Solomon) and may inspire further cross‑fertilization between additive combinatorics and coding theory.

In summary, the authors provide a novel, combinatorial‑information‑theoretic proof that Reed–Muller codes achieve Shannon capacity in the weak sense. Their approach avoids the heavy analytic machinery traditionally used for polar codes, instead leveraging recent advances in additive combinatorics (the PFR theorem) and a new orbit‑localization lemma to control conditional entropies of code layers. This work not only settles a long‑standing open problem about RM codes but also opens new avenues for applying additive combinatorial techniques to the analysis of deterministic capacity‑achieving codes.