Compression of Periodic Complementary Sequences and Applications

TL;DR

This paper introduces a compression method for complex sequences that preserves their complementary properties, simplifying the construction of such sequences and resolving many open existence questions for sequences of length up to 50.

Contribution

It proves that m-compression preserves complementarity and uses this to construct new complementary sequences and difference sets, solving most open cases for lengths up to 50.

Findings

m-compression preserves complementarity

Constructed new complementary sequences of composite lengths

Resolved most open existence cases for sequences up to length 50

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,...,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} + ... + 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 {+1,-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 <= 50 for which the existence question was open. We resolve all but one of these cases.