Periodic complementary sets of binary sequences

TL;DR

This paper constructs new cyclic difference families to completely solve the problem of existence for periodic complementary sets of binary sequences, advancing the understanding of their structure and existence conditions.

Contribution

The paper introduces new cyclic difference families that fully resolve the existence problem for periodic complementary sets of binary sequences.

Findings

Complete solution to the existence problem for PCS_p^N.

Construction of new cyclic difference families.

Resolution of previously undecided cases.

Abstract

Let PCS_p^N denote a set of p binary sequences of length N such that the sum of their periodic auto-correlation functions is a delta-function. In the 1990, Boemer and Antweiler addressed the problem of constructing such sequences. They presented a table covering the range p <= 12, N <= 50 and showing in which cases it was known at that time whether such sequences exist, do not exist, or the question of existence is undecided. The number of undecided cases was rather large. Subsequently the number of undecided cases was reduced to 26 by the author. In the present note, several cyclic difference families are constructed and used to obtain new sets of periodic binary sequences. Thereby the original problem of Boemer and Antweiler is completely solved.