Golay's Merit Factor

The problem

For $n\ge 1$, let ${\mathbb{U}}_n$ denote the set of polynomials $p(z)$ of degree $n$ with coefficients $\pm 1$. Define $$C^{-}(n):=\underset{p\in {\mathbb{U}}_n}{\max }\left(\underset{|z|=1}{\min }\frac{|p(z)|}{\sqrt{n+1}}\right)$$ $$C^{+}(n):=\underset{p\in {\mathbb{U}}_n}{\min }\left(\underset{|z|=1}{\max }\frac{|p(z)|}{\sqrt{n+1}}\right)$$ $$C^w(n):=\underset{p\in {\mathbb{U}}_n}{\min }\left(\underset{|z|=1}{\max }\frac{|p(z)|}{\sqrt{n+1}}-\underset{|z|=1}{\min }\frac{|p(z)|}{\sqrt{n+1}}\right)$$ $$C^4(n):=\underset{p\in {\mathbb{U}}_n}{\min }\frac{(n+1{)}^2}{{\int }_0^1|p(e^{2\pi \theta }){|}^{4}d\theta -(n+1{)}^2}$$ (The quantity being minimized for $C^4(n)$ is known as Golay's merit factor for $p$.) What is the behavior of $C^{-}(n)$, $C^{+}(n)$, $C^w(n)$, $C^4(n)$ as $n\to \infty$?

AlphaEvolve's best construction is below the known record.

What counts as progress

• A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
• A proof that a known construction is optimal, or a better bound on the other side.
• Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.