The Zero-Difference Properties of Functions and Their Applications
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| Format: | Preprint |
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2018
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| _version_ | 1866914225239097344 |
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| author | Yi, Zongxiang Pei, Dingyi ChunmingTang |
| author_facet | Yi, Zongxiang Pei, Dingyi ChunmingTang |
| contents | A function $f$ from an Abelian group $(A,+)$ to an Abelian group $(B,+)$ is $(n, m, S)$ zero-difference (ZD), if $S=\{λ_α\mid α\in A\setminus\{0\}\}$ where $n=|A|$, $m=|f(A)|$ and $λ_α=|\{x \in A \mid f(x+α)=f(x)\}|$. A function is called zero-difference balanced (ZDB) if $S=\{λ\}$ where $λ$ is a constant number. ZDB functions have many good applications. However it is point out that many known zero-difference balanced functions are already given in the language of partitioned difference family (PDF). The problem that whether zero-difference ``not balanced" functions still have good applications as ZDB functions, is investigated in this paper. By using the change point technic, zero-difference functions with good applications are constructed from known ZDB functions. Then optimal difference systems of sets (DSS) and optimal frequency-hopping sequences (FHS) are obtained with new parameters. Furthermore the sufficient and necessary conditions of these objects being optimal, are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_08132 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The Zero-Difference Properties of Functions and Their Applications Yi, Zongxiang Pei, Dingyi ChunmingTang Combinatorics Cryptography and Security A function $f$ from an Abelian group $(A,+)$ to an Abelian group $(B,+)$ is $(n, m, S)$ zero-difference (ZD), if $S=\{λ_α\mid α\in A\setminus\{0\}\}$ where $n=|A|$, $m=|f(A)|$ and $λ_α=|\{x \in A \mid f(x+α)=f(x)\}|$. A function is called zero-difference balanced (ZDB) if $S=\{λ\}$ where $λ$ is a constant number. ZDB functions have many good applications. However it is point out that many known zero-difference balanced functions are already given in the language of partitioned difference family (PDF). The problem that whether zero-difference ``not balanced" functions still have good applications as ZDB functions, is investigated in this paper. By using the change point technic, zero-difference functions with good applications are constructed from known ZDB functions. Then optimal difference systems of sets (DSS) and optimal frequency-hopping sequences (FHS) are obtained with new parameters. Furthermore the sufficient and necessary conditions of these objects being optimal, are given. |
| title | The Zero-Difference Properties of Functions and Their Applications |
| topic | Combinatorics Cryptography and Security |
| url | https://arxiv.org/abs/1811.08132 |