Structural properties of Costas arrays
Jonathan Jedwab Jane Wodlinger
We apply combinatorial arguments to establish structural constraints on Costas arrays. We prove restrictions on when a Costas array can contain a large corner region whose entries are all 0. In particular, we prove a 2010 conjecture due to Russo, Erickson and Beard. We then constrain the vectors joining pairs of 1s in a Costas array by establishing a series of results on its number of "mirror pairs," namely pairs of these vectors having the same length but opposite slopes.
keywords: Costas array mirror pair structural constraints. common vector all-zero corner region
Bounds on the growth rate of the peak sidelobe level of binary sequences
Denis Dmitriev Jonathan Jedwab
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.
keywords: asymptotic. peak sidelobe level maximal length shift register sequence binary sequence Aperiodic autocorrelation $m$-sequence
Two binary sequence families with large merit factor
Kai-Uwe Schmidt Jonathan Jedwab Matthew G. Parker
We calculate the asymptotic merit factor, under all rotations of sequence elements, of two families of binary sequences derived from Legendre sequences. The rotation is negaperiodic for the first family, and periodic for the second family. In both cases the maximum asymptotic merit factor is 6. As a consequence, we obtain the first two families of skew-symmetric sequences with known asymptotic merit factor, which is also 6 in both cases.
keywords: skew-symmetric. negaperiodic construction asymptotic merit factor rotation Legendre sequence periodic Binary sequence

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