Compression of periodic complementary sequences and applications

Abstract

A collection of complex sequences of length $v$ is complementary if the sum of their periodic autocorrelation function values at all non-zero shifts is constant. For a complex sequence $A = [a_0,a_1,\ldots ,a_{v-1}]$ of length $v = dm$ we define the $m$-compressed sequence $A^{(d)}$ of length $d$ whose terms are the sums $a_i + a_{i+d} + \cdots + a_{i+(m-1)d}$. We prove that the $m$-compression of a complementary collection of sequences is also complementary. The compression procedure can be used to simplify the construction of complementary $\{\pm 1\}$-sequences of composite length. In particular, we construct several supplementary difference sets $(v;r,s;\lambda )$ with $v$ even and $\lambda = (r+s)-v/2$, given here for the first time. There are 15 normalized parameter sets $(v;r,s;\lambda )$ with $v\le 50$ for which the existence question was open. We resolve all but one of these cases.