新的Ⅱ型偶长Z-互补对构造

doi:10.11959/j.issn.1000-436x.2023052

Abstract

Abstract:

To enrich the parameter forms of Z-complementary pair (ZCP), the constructions of new Type-II ZCP were investigated.First, Type-II $(2^{k + 2} N, (2^{k + 2} - 1) N) - ZCP$ was proposed by using iterative method.When N= 1, Z-optimal type-II $(2^{k + 2}, (2^{k + 2} - 1)) - ZCP$ could be obtained.Then, type-II $(2^{k + 1} N + 2^{k + 1}, 2^{k + 1} N + 2^{k + 1} - \frac{N}{2}) - ZCP$ was presented by combining the iterative method with the insertion method.When N= 2, Z-optimal type-II $(3 \times 2^{k + 1}, 3 \times 2^{k + 1} - 1) - ZCP$ could be obtained.The proposed constructions can generate new even-length ZCP, and expand the number of existing ZCP.

Key words: aperiodic, iterative method, Golay complementary pair, Z-complementary pair

CLC Number:

TN911.2

Xiaoyu CHEN, Lianfeng SUN, Jiekun ZHANG, Yubo LI. Construction of new even-length type-II Z-complementary pair[J]. Journal on Communications, 2023, 44(3): 138-144.

Figures/Tables 5

长度	$(a, b)$
2	$(\begin{matrix} + + \\ + - \end{matrix})$
10	$(\begin{matrix} + + - + - + - - + + \\ + + - + + + + + - - \end{matrix})$
26	$(\begin{matrix} + + + + - + + - - + - + - + - - + - + + + - - + + + \\ + + + + - + + - - + - + + + + + - + - - - + + - - - \end{matrix})$

长度	ZCZ宽度	$ZCP (s, t)$	${\| ρ_{s} (τ) + ρ_{t} (τ) \|}_{τ = 0}^{N - 1}$
4	3	$(\begin{matrix} + + - - \\ + + + + \end{matrix})$	$(8, 4, 0_{2})$
8	7	$(\begin{matrix} + + - - + + + + \\ + + - - - - - - \end{matrix})$	$(16, 8, 0_{6})$
16	15	$(\begin{matrix} + + - - + + + + + + - - - - - - \\ + + - - + + + + - - + + + + + + \end{matrix})$	$(32, 16, 0_{14})$
32	31	$(\begin{matrix} + + - - + + + + + + - - - - - - + + - - + + + + - - + + + + + + \\ + + - - + + + + + + - - - - - - - - + + - - - - + + - - - - - - \end{matrix})$	$(64, 32, 0_{30})$

构造方法	长度	ZCZ宽度	$ZCP (s, t)$	${\| ρ_{s} (τ) + ρ_{t} (τ) \|}_{τ = 0}^{N - 1}$
文献[16]	6	4	$(\begin{matrix} + - + + - + \\ - + - + - + \end{matrix})$	$(12, 8, 4, 0_{3})$
	24	22	$(\begin{matrix} + - + + - + + - + - + - + - + + - + - + - + - + \\ - + - + - + - + - - + - - + - + - + + - + + - + \end{matrix})$	$(48, 32, 16, 0_{21})$
本文定理2	6	5	$(\begin{matrix} + + + + + - \\ + + + - - + \end{matrix})$	$(12, 4, 0_{4})$
本文定理3	24	23	$(\begin{matrix} + + + + + - + + + - - + + + + + + - - - - + + - \\ + + + + + - + + + - - + - - - - - + + + + - - + \end{matrix})$	$(48, 16, 0_{22})$

构造方法	ZCP长度	ZCZ宽度	零相关区外的值(v)	ZCZ比率	是否Z-最优
文献[15]定理2	$\begin{array}{l} 2^{k} N + 2^{k - 1} \\ N \in ℤ^{+}, k \in {2, 3, 4 \dots} \end{array}$	2^kN+2^k-1-N	4≤v≤2^k(2N-1)	$\frac{2^{k} - 1}{2^{k}}$	当N=1时，Z-最优
文献[15]定理3	$3 N, N = 2^{α} 10^{β} 26^{γ}$	3N-1	2N	1	是
	$14 N, N = 2^{α} 10^{β} 26^{γ}$	14N-1	4N	1	是
本文定理1	$\begin{array}{l} 2^{k + 2} N \\ k \in {1, 2, 3, \dots} \end{array}$	(2^{k+ 2}-1)N	2^k × 4N	$\frac{2^{k + 2} - 1}{2^{k + 2}}$	当N=1时，Z-最优
	$N = 2^{α} 10^{β} 26^{γ}$
本文定理2	$\begin{array}{l} 2 N + 2 \\ N = 2^{α} 10^{β} 26^{γ} \end{array}$	$\frac{3 N}{2} + 2$	4	$\frac{3}{4}$	当N=2时，Z-最优
本文定理3	$\begin{array}{l} 2^{k + 1} N + 2^{k + 1} \\ k \in {1, 2, 3, \dots} \end{array}$	$2^{k + 1} N + 2^{k + 1} - \frac{N}{2}$	2^{k+ 2}	$\frac{2^{k + 2} - 1}{2^{k + 2}}$	当N=2时，Z-最优
	$N = 2^{α} 10^{β} 26^{γ}$

构造方法	长度	ZCZ宽度	零相关区外的值	${\| ρ_{e_{i}} (τ) + ρ_{f_{i}} (τ) \|}_{τ = 0}^{N - 1}, i \in {0, 1}$	PMEPR
	10	8	4≤v≤12	$(20, 12, 4, 0_{7})$	(4.4 444,3.600 0)
文献[15]定理2	18	14	4≤v≤28	$(36, 4, 20, 4, 4, 0_{13})$	(4.3 622,3.410 6)
	34	26	4≤v≤60	$(68, 20, 20, 12, 4, 4, 12, 12, 4, 0_{25})$	(4.2 141,4.446 5)
	10	8	4	$(20, 4, 4, 0_{7})$	(3.6 000,2.868 1)
本文定理2	18	14	4	$(36, 4, 4, 4, 4, 0_{13})$	(2.5 060,2.711 8)
	34	26	4	$(68, 4, 4, 4, 4, 4, 4, 4, 4, 0_{25})$	(2.9 412,2.521 6)

References 17

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