NEARLY PERFECT SEQUENCES WITH ARBITRARY OUT-OF-PHASE AUTOCORRELATION


YAYLA O.

ADVANCES IN MATHEMATICS OF COMMUNICATIONS, vol.10, no.2, pp.401-411, 2016 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 10 Issue: 2
  • Publication Date: 2016
  • Doi Number: 10.3934/amc.2016014
  • Title of Journal : ADVANCES IN MATHEMATICS OF COMMUNICATIONS
  • Page Numbers: pp.401-411
  • Keywords: Perfect sequence, nearly perfect sequence, direct product difference set, relative difference set, multiplier

Abstract

A sequence of period n is called a nearly perfect sequence of type gamma if all out-of-phase autocorrelation coefficients are a constant gamma. In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a p-ary NPS of period n and type gamma and a cyclic (n,p,n, n-gamma/p + gamma, 0, n-gamma/p)-DPDS for an arbitrary integer gamma. Next, we present the necessary conditions for the existence of a p-ary NPS of type gamma. We apply this result for excluding the existence of some p-ary NPS of period n and type gamma for n <= 100 and vertical bar gamma vertical bar <= 2. We also prove the similar results for an almost p-ary NPS of type gamma. Finally, we show the non-existence of some almost p-ary perfect sequences by showing the non-existence of equivalent cyclic relative difference sets by using the notion of multipliers.