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Pursley and Sarwate established a lower bound on a combined measure ofautocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences(i.e., sequences with terms in $\{-1,1\}$). If $f$ is a nonzero sequence, thenits autocorrelation demerit factor, $\text{ADF}(f)$, is the sum of the squaredmagnitudes of the aperiodic autocorrelation values over all nonzero shifts forthe sequence obtained by normalizing $f$ to have unit Euclidean norm. If$(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demeritfactor, $\text{CDF}(f,g)$, is the sum of the squared magnitudes of theaperiodic crosscorrelation values over all shifts for the sequences obtained bynormalizing both $f$ and $g$ to have unit Euclidean norm. Pursley and Sarwateshowed that for binary sequences, the sum of $\text{CDF}(f,g)$ and thegeometric mean of $\text{ADF}(f)$ and $\text{ADF}{(g)}$ must be at least $1$.For randomly selected pairs of long binary sequences, this quantity istypically around $2$. In this paper, we show that Pursley and Sarwate's boundis met for binary sequences precisely when $(f,g)$ is a Golay complementarypair. We also prove a generalization of this result for sequences whose termsare arbitrary complex numbers. We investigate constructions that produceinfinite families of Golay complementary pairs, and compute the asymptoticvalues of autocorrelation and crosscorrelation demerit factors for suchfamilies.

Sequence Pairs with Lowest Combined Autocorrelation and Crosscorrelation