Sets of Low Correlation Sequences from Cyclotomy

Abstract

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.