A Computer Search of New OBZCPs of Lengths up to 49

TL;DR

This article introduces a computer search algorithm with time complexity and presents a table of OBZCPs with smallest combined auto-correlation and cross-correlation, based on the Pursley–Sarwate criterion.

Abstract

This paper aims to search for new optimal and sub-optimal Odd Binary Z-Complimentary Pairs (OBZCPs) for lengths up to 49. As an alternative to the celebrated binary Golay complementary pairs, optimal OBZCPs are the best almost-complementary sequence pairs having odd lengths. We introduce a computer search algorithm with time complexity $O(2^N)$, where $N$ denotes the sequence length and then show optimal results for all $27 \le N \le 33$ and $N=37,41,49$. For those sequence lengths (i.e., $N=35,39,43,45,47$) with no optimal pairs, we show OBZCPs with largest zero-correlation zone (ZCZ) widths (i.e., $Z$-optimal). Finally, based on the Pursley--Sarwate criterion (PSC), we present a table of OBZCPs with smallest combined auto-correlation and cross-correlation.

Figures (2)

• Figure 1: Structure of sequence $\mathbf{a}$ of length 7.
• Figure 2: Plots of AACF sums for OBZCPs, i.e., (7905A9444, 710C1A3B2) and (15BCD1FAF3340,10599EA0E984A), of lengths $35$ and $49$, respectively.

Theorems & Definitions (3)

• Definition 1
• Example 1
• Example 2