Partial direct product difference sets and sequences with ideal autocorrelation

TL;DR

This paper explores sequences with two zero-symbols and ideal autocorrelation, establishing their equivalence to partial direct product difference sets and providing new construction methods using cyclotomic classes.

Contribution

It extends the theory of sequences with zero-symbols to non-consecutive cases and introduces a construction method for almost quaternary sequences with ideal autocorrelation.

Findings

Sequences with two zero-symbols are equivalent to $ ext{ell}$-partial direct product difference sets.

Extended results from consecutive to non-consecutive zero-symbols.

Constructed a family of almost quaternary sequences with ideal autocorrelation using cyclotomic classes.

Abstract

In this paper, we study the sequences with (non-consecutive) two zero-symbols and ideal autocorrelation, which are also known as almost $m$-ary nearly perfect sequences. We show that these sequences are equivalent to $\ell$-partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for $\ell$-PDPDS. Finally, we present a construction method for a family of almost quaternary sequences with ideal autocorrelation by using cyclotomic classes.