  Low correlation (finite length) sequences are used in communications and
remote sensing. One seeks codebooks of sequences in which each sequence has low
aperiodic autocorrelation at all nonzero shifts, and each pair of distinct
sequences has low aperiodic crosscorrelation at all shifts. An overall
criterion of codebook quality is the demerit factor, which normalizes all
sequences to unit Euclidean norm, sums the squared magnitudes of all the
correlations between every pair of sequences in the codebook (including
sequences with themselves to cover autocorrelations), and divides by the square
of the number of sequences in the codebook. This demerit factor is expected to
be $1+1/N-1/(\ell N)$ for a codebook of $N$ randomly selected binary sequences
of length $\ell$, but we want demerit factors much closer to the absolute
minimum value of $1$. For each $N$ such that there is an $N\times N$ Hadamard
matrix, we use cyclotomy to construct an infinite family of codebooks of binary
sequences, in which each codebook has $N-1$ sequences of length $p$, where $p$
runs through the primes with $N\mid p-1$. As $p$ tends to infinity, the demerit
factor of the codebooks tends to $1+1/(6(N-1))$, and the maximum magnitude of
the undesirable correlations (crosscorrelations between distinct sequences and
off-peak autocorrelations) is less than a small constant times
$\sqrt{p}\log(p)$. This construction also generalizes to nonbinary sequences.
