A new construction of zero-difference balanced functions and two applications

Liu, Junying; Jiang, Yupeng; Zheng, Qun-Xiong; Lin, Dongdai

doi:10.1007/s10623-019-00616-x

Cited by 3 publications

(7 citation statements)

References 23 publications

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“…(2) In this paper, we present three constructions of PDFs based on the generalized cyclotomy above and d-form functions with difference balanced property, see Theorems 3.4, 3.6 and 3.9. Firstly, compared with Construction 1 in [6] and Construction 1 in [29] respectively, Construction 1 and Construction 2 in this paper provide new parameters since the requirement e|(p mi i − 1) gives more flexibility. Furthermore, compared with the recursive constructions of PDFs in [27,Theorem 18] and [4,Chapter 3], our constructions are direct.…”

Section: Lemma 11 [42]mentioning

confidence: 99%

“…= (33,24,24,24,25,1,2,3,26,5,6,7,27,9,10,11,28,13,14,15,29,17,18,19,30,21,22,23,31,2,3,0,32,6,7,4,25,10,11,8,26,14,15,12,27,18,19,16,28,22,23,20,29,3,…”

Section: Lemma 31 Let An Integermentioning

confidence: 99%

“…More recently, the cyclotomy has proved to be valuable in other applied fields such as sequences [40,15,8,22,9], coding theory [15,17,16,10,25] and cryptography [15]. The combinatorics has also benefited from the use of cyclotomy, which can be applied for constructing difference sets, difference families, and so on [32,35,17,41,36,7,39,34,37,6,29].…”

Section: Introductionmentioning

confidence: 99%

“…There are various methods of constructing PDFs, which have been presented in [36,7,39,34,14,18,20,5,27,2,3,4]. Recently, PDFs have been investigated intensively under the notion of zero-difference balanced function [41,37,6,29,11,12,42,33,38,13]. Let f be a function from an additive group A onto a set B, where |A| = n and |B| = m. f is called an (n, m, λ) zero-difference balanced (ZDB) function if for any nonzero a ∈ A, we have |{x ∈ A : f (a + x) = f (x)}| = λ for some constant λ.…”

Section: Introductionmentioning

confidence: 99%

“…Let f be a function from an additive group A onto a set B, where |A| = n and |B| = m. f is called an (n, m, λ) zero-difference balanced (ZDB) function if for any nonzero a ∈ A, we have |{x ∈ A : f (a + x) = f (x)}| = λ for some constant λ. In recent years, it was proved that ZDB functions have played an important role in the constructions of optimal constant composition codes [41,37,6,29,14,11,42,38], optimal constant weight codes [42,33,38], optimal frequency-hopping sequences [6,18,20,33,38], optimal and perfect difference systems of sets [41,6,29,12,42,38]. Due to the widespread applications of ZDB functions, many researchers have been occupied in constructing ZDB functions.…”

Section: Introductionmentioning

confidence: 99%

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Three classes of partitioned difference families and their optimal constant composition codes

Xu¹,

Qu²,

Cao³

2022

AMC

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<p style='text-indent:20px;'>Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula> prime or composite, cyclotomy on a residue class ring <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb{Z}}_{v} $\end{document}</tex-math></inline-formula> can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [<xref ref-type="bibr" rid="b40">40</xref>], we introduce a generalized cyclotomy of order <inline-formula><tex-math id="M4">\begin{document}$ e $\end{document}</tex-math></inline-formula> on the ring <inline-formula><tex-math id="M5">\begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q_j $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ i\neq j $\end{document}</tex-math></inline-formula>) may not be co-prime, which includes classical cyclotomy as a special case. Here, <inline-formula><tex-math id="M9">\begin{document}$ q_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ q_k $\end{document}</tex-math></inline-formula> are powers of primes with an integer <inline-formula><tex-math id="M13">\begin{document}$ e|(q_i-1) $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M14">\begin{document}$ 1\leq i\leq k $\end{document}</tex-math></inline-formula>. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and <inline-formula><tex-math id="M15">\begin{document}$ d $\end{document}</tex-math></inline-formula>-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.</p>

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Section: Lemma 11 [42]mentioning

confidence: 99%

“…= (33,24,24,24,25,1,2,3,26,5,6,7,27,9,10,11,28,13,14,15,29,17,18,19,30,21,22,23,31,2,3,0,32,6,7,4,25,10,11,8,26,14,15,12,27,18,19,16,28,22,23,20,29,3,…”

Section: Lemma 31 Let An Integermentioning

confidence: 99%