Partial direct product difference sets and almost quaternary sequences

Büşra Özden; Oǧuz Yayla

doi:10.3934/amc.2021010

Article Contents

2023, Volume 17, Issue 3: 572-588. Doi: 10.3934/amc.2021010

This issue Previous Article Formal security proof for a scheme on a topological network Next Article The weight recursions for the 2-rotation symmetric quartic Boolean functions

Partial direct product difference sets and almost quaternary sequences

Büşra Özden^1, , and
Oǧuz Yayla^2,

1.
Hacettepe University, Graduate School of Science and Engineering, Beytepe, Ankara, Turkey
2.
Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

^* Corresponding author
^* Corresponding author

This paper is a part of Büşra Özden's PhD thesis

Received: May 2020

Revised: January 2021

Early access: May 2021

Published: June 2023

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Abstract

In this paper, we study the $m$ -ary sequences with (non-consecutive) two zero-symbols and at most two distinct autocorrelation coefficients, which are 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 two construction methods for a family of almost quaternary sequences with at most two out-of-phase autocorrelation coefficients.

Keywords:

Mathematics Subject Classification: 05B10 and 94A55.

Citation:

Full Text(HTML)

Table 1. Orbits of $G = {\mathbb Z}_{10} \times {\mathbb Z}_3$ under $x \rightarrow 19x$

orbits of length 1
(0, 0)}, {(0, 1)}, {(0, 2)}, {(5, 0)}, {(5, 1)}, {(5, 2)
orbits of length 2
(4, 0), (6, 0)}, {(1, 1), (9, 1)}, {(7, 0), (3, 0)}, {(8, 2), (2, 2)},
(9, 0), (1, 0)} {(7, 2), (3, 2)}, {(7, 1), (3, 1)}, {(1, 2), (9, 2)},
(8, 1), (2, 1)}, {(6, 2), (4, 2)}, {(6, 1), (4, 1)}, {(2, 0), (8, 0)

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Table 2. Sequences, their autocorrelation and alphabet

Construction	Out-of-phase autocorrelation	Alphabet
[29], [36]	0	$p$ -ary
[17], [18]	-1	$p$ -ary
[3], [10], [15], [16], [30], [32], [35], [41], [42], [43], [44]	-1	binary
[12], [27], [40], [44]	$\pm 2$	binary
[2], [40], [27], [44]	$(0,-4)$	binary
[45], [50]	$(0,\pm 4)$	binary
[4], [11], [44]	$(1,-3)$	binary
[25]	$(2p,-2)$ or $(\pm 2p, \pm 2)$	binary
[1], [19], [21]	-1	ternary
[23]	$(0,-3,3\zeta_3,3\zeta_3^2)$	ternary
	$(0,\pm 2i,-4,-2,-2\pm 2i)$ or $(0,\pm 2i,\pm2,-2\pm 2i)$	quaternary
[24]	$(-2,\pm 2i)$	quaternary
[25], [44]	$(0,-2)$	quaternary
[31]	$(-1,\pm 3)$	quaternary
[46]	$(-1,\pm(1+2i))$ or $(\pm 1,-3)$	quaternary
[48]	$(\frac{p^{n-1}(p-7)}{2},\frac{p^{n-1}(p-3)}{2},p^n)$	quaternary
[6]	$0$	$p$ -ary with one zero
	$-1$	$p$ -ary with one zero
[38]	$-1$	$m$ -ary with one zero
[13]	$(0,3q^{n-1})$	$6$ -ary with one zero
[47]	$(0,p^{(k-1)n})$	$p^{kn}$ -ary with zeros
	$(0,p^{(k-1)n})$	$\frac{p^{n-1}}{\gcd(t,p^{n-1})}$ -ary with zeros
	$0$	$m$ -ary with one zero
Theorem 3.7	$\frac{q-3}{2}$	quaternary with one zero
Proposition 7	$(-1,0)$	$m$ -ary with $\frac{q}{2}+1$ zeros

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