The peak sidelobe level of random binary sequences
Copyright policy )The peak sidelobe level of random binary sequences
Let $A_n=(a_0,a_1,\dots,a_{n-1})$ be drawn uniformly at random from $\{-1,+1\}^n$ and define \[ M(A_n)=\max_{01$}. \] It is proved that $M(A_n)/\sqrt{n\log n}$ converges in probability to $\sqrt{2}$. This settles a problem first studied by Moon and Moser in the 1960s and proves in the affirmative a recent conjecture due to Alon, Litsyn, and Shpunt. It is also shown that the expectation of $M(A_n)/\sqrt{n\log n}$ tends to $\sqrt{2}$.
minor revisions and corrections compared to the first version
- Otto-von-Guericke University Magdeburg Germany
- Simon Fraser University Canada
FOS: Computer and information sciences, Computer Science - Information Theory, Information Theory (cs.IT), Shift register sequences and sequences over finite alphabets in information and communication theory, Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.), Large deviations, aperiodic autocorrelation, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), 05D40 (Primary) 94A55, 60F10 (Secondary)
FOS: Computer and information sciences, Computer Science - Information Theory, Information Theory (cs.IT), Shift register sequences and sequences over finite alphabets in information and communication theory, Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.), Large deviations, aperiodic autocorrelation, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), 05D40 (Primary) 94A55, 60F10 (Secondary)
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