# American Institute of Mathematical Sciences

November  2007, 1(4): 461-475. doi: 10.3934/amc.2007.1.461

## Bounds on the growth rate of the peak sidelobe level of binary sequences

 1 The D. E. Shaw Group, 39th Floor, Tower 45, 120 West Forty-Fifth Street, New York, NY 10036, United States 2 Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6, Canada

Received  May 2007 Revised  October 2007 Published  October 2007

The peak sidelobe level (PSL) of a binary sequence is the largest absolute value of all its nontrivial aperiodic autocorrelations. A classical problem of digital sequence design is to determine how slowly the PSL of a length $n$ binary sequence can grow, as $n$ becomes large. Moon and Moser showed in 1968 that the growth rate of the PSL of almost all length $n$ binary sequences lies between order $\sqrt{n\log n}$ and $\sqrt{n}$, but since then no theoretical improvement to these bounds has been found.
We present the first numerical evidence on the tightness of these bounds, showing that the PSL of almost all binary sequences of length $n$ appears to grow exactly like order $\sqrt{n\log n}$, and that the PSL of almost all $m$-sequences of length $n$ appears to grow exactly like order $\sqrt{n}$. In the case of $m$-sequences, a key algorithmic insight reveals behaviour that was previously well beyond the range of computation.
Citation: Denis Dmitriev, Jonathan Jedwab. Bounds on the growth rate of the peak sidelobe level of binary sequences. Advances in Mathematics of Communications, 2007, 1 (4) : 461-475. doi: 10.3934/amc.2007.1.461
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