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Almost perfect sequences with $θ=2$. (English)
Arch. Math. 70, No.2, 128-131 (1998).
An almost perfect sequence is a periodic $\pm 1$-sequence where all the off-peak autocorrelation coefficients are as small as theoretically possible, with exactly one exception, say $C(g)$. It is easy to see that the period $v$ of the sequence has to be even, say $v= 2m$, and that $g= m$. By a result of {\it A. Pott} and {\it S. P. Bradley} [Existence and nonexistence of almost perfect autocorrelation sequences, IEEE Trans. Inf. Theory 41, No. 1, 301-304 (1995; Zbl 0830.05011)], the subset of ${\bbfZ}_{2m}$ corresponding to the $k$ (say) entires $+1$ in the first period of an almost perfect sequence is a divisible difference set with parameters $(m,2,k,(k- m)(k- m+1), k-(m/2))$. The authors study the case $θ= 2$, where $θ:= k-m$; thus such an almost perfect sequence is equivalent to a divisible difference set in ${\bbfZ}_{2m}$ with parameters $(m,2,m-2, 2, (m-4)/2)$. Building on a previous paper by {\it K. T. Arasu}, {\it S. L. Ma} and {\it N. J. Voss} [On a class of almost perfect sequences, J. Algebra 192, No. 2, 641-650, Art. No. JA976962 (1997; Zbl 0880.05019)], they show that these objects exist if and only if $m\in\{8, 12, 28\}$.
Reviewer: D.Jungnickel (Augsburg)