AlphaEvolve problems·alphaevolve:28

Golay's Merit Factor

For $n \\geq 1$, let $\\mathbb{U}_{n}$ denote the set of polynomials $p(z)$ of degree $n$ with coefficients $\\pm 1$. Define $$ C^-(n) \\coloneqq \\max_{p \\in \\mathbb{U}_{n}} \\left(\min_{|z|=1}\\frac{|p(z)|}{\\sqrt{n+1}}\\right) $$ $$ C^+(n) \\coloneqq \\min_{p \\in \\mathbb{U}_{n}}\\left(\\max_{|z|=1}\\frac{|p(z)|}{\\sqrt{n+1}}\\right) $$ $$ C^w(n) \\coloneqq \\min_{p \\in \\mathbb{U}_{n}}\\left(\\max_{|z|=1}\\frac{|p(z)|}{\\sqrt{n+1}} - \\min_{|z|=1}\\frac{|p(z)|}{\\sqrt{n+1}}\\right) $$ $$ C^4(n) \\coloneqq \\min_{p \\in \\mathbb{U}_{n}} \\frac{(n+1)^2}{\\int_0^1 |p(e^{2\\pi \\theta})|^4\ d\\theta - (n+1)^2} $$ (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 $n \\to \\infty$?

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