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comment-literature: The merit factor convention and the Legendre sequence family are those studied by Golay CITE{Golay} and by Hoholdt and Jensen CITE{HJ}.- comment-fekete: If $U_p(x)=\sum_{j=0}^{p-1}u_jx^j$, then $U_p(x)=1+f_p(x)$, where- $f_p$ is the HREF{Fekete_polynomials}[Fekete polynomial] with the same prime parameter.+ comment-fekete: The polynomial $U_p$ from CITE{formula-l4-norm} satisfies $U_p(x)=1+f_p(x)$,+ where $f_p$ is the HREF{Fekete_polynomials}[Fekete polynomial] with the same prime+ parameter. Formulas: formula-merit-factor: $F_p=p^2/(2\sum_{k=1}^{p-1}c_k^2)$, where $c_k=\sum_{j=0}^{p-1-k}u_ju_{j+k}$- and $c_0=p$.- formula-l4-norm: If $\|U_p\|_4^4$ denotes the integral of $|U_p(z)|^4$ over $|z|=1$- with normalized Haar measure, then $F_p=p^2/(\|U_p\|_4^4-p^2)$.+ for $1\leq k<p$.+ formula-l4-norm: If $U_p(x)=\sum_{j=0}^{p-1}u_jx^j$ and $\|U_p\|_4^4$ denotes the+ integral of $|U_p(z)|^4$ over $|z|=1$ with normalized Haar measure, then $F_p=p^2/(\|U_p\|_4^4-p^2)$,+ since $\|U_p\|_4^4=c_0^2+2\sum_{k=1}^{p-1}c_k^2$ with $c_0=\sum_{j=0}^{p-1}u_j^2=p$. formula-asymptotic: For the unrotated Legendre sequence, $F_p\to 3/2$ as $p\to\infty$ CITE{HJ}.
- table: HREF{Complementary_Shapiro_polynomials}[Complementary Shapiro polynomials $Q_n$]- relation: store the second polynomial in the Rudin-Shapiro recursion rather than- a Legendre-symbol sequence merit factor+ relation: form the second polynomial sequence in the same Rudin-Shapiro pair, another+ family of binary sequences with small aperiodic autocorrelations Links: Legendre:
rigour: exact complete: 'no'- complete-note: it holds every odd prime $p<1000$+ complete-note: it holds every odd prime $p<1000$, a small-prime range that keeps+ the exact $O(p^2)$ autocorrelation computation tractable to rerun rigour details: The generator computes the Legendre sequence from exact Legendre symbols and forms the aperiodic autocorrelations by integer arithmetic. The values
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