Walsh spectrum and nega spectrum of complementary arrays
Abstract It has been shown that all the known binary Golay complementary sequences of length $$2^m$$ can be obtained by a single binary Golay complementary array of dimension m and size $$2\times 2 \times \cdots \times 2$$ which can be represented by a Boolean function. However, the construction of...
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| Autor*in: |
Chai, Jinjin [verfasserIn] |
|---|
| Format: |
Artikel |
|---|---|
| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Designs, codes and cryptography - Springer US, 1991, 89(2021), 12 vom: 19. Okt., Seite 2663-2677 |
|---|---|
| Übergeordnetes Werk: |
volume:89 ; year:2021 ; number:12 ; day:19 ; month:10 ; pages:2663-2677 |
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| DOI / URN: |
10.1007/s10623-021-00938-9 |
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| 520 | |a Abstract It has been shown that all the known binary Golay complementary sequences of length $$2^m$$ can be obtained by a single binary Golay complementary array of dimension m and size $$2\times 2 \times \cdots \times 2$$ which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length $$2^m$$ or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions. | ||
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10.1007/s10623-021-00938-9 doi (DE-627)OLC2077352531 (DE-He213)s10623-021-00938-9-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Chai, Jinjin verfasserin aut Walsh spectrum and nega spectrum of complementary arrays 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract It has been shown that all the known binary Golay complementary sequences of length $$2^m$$ can be obtained by a single binary Golay complementary array of dimension m and size $$2\times 2 \times \cdots \times 2$$ which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length $$2^m$$ or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions. Complementary array Walsh–Hadamard transform Nega–Hadamard transform Boolean function Wang, Zilong (orcid)0000-0002-1525-3356 aut Xue, Erzhong aut Enthalten in Designs, codes and cryptography Springer US, 1991 89(2021), 12 vom: 19. Okt., Seite 2663-2677 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:89 year:2021 number:12 day:19 month:10 pages:2663-2677 https://doi.org/10.1007/s10623-021-00938-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 89 2021 12 19 10 2663-2677 |
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10.1007/s10623-021-00938-9 doi (DE-627)OLC2077352531 (DE-He213)s10623-021-00938-9-p DE-627 ger DE-627 rakwb eng 004 VZ 17,1 ssgn Chai, Jinjin verfasserin aut Walsh spectrum and nega spectrum of complementary arrays 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 Abstract It has been shown that all the known binary Golay complementary sequences of length $$2^m$$ can be obtained by a single binary Golay complementary array of dimension m and size $$2\times 2 \times \cdots \times 2$$ which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length $$2^m$$ or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions. Complementary array Walsh–Hadamard transform Nega–Hadamard transform Boolean function Wang, Zilong (orcid)0000-0002-1525-3356 aut Xue, Erzhong aut Enthalten in Designs, codes and cryptography Springer US, 1991 89(2021), 12 vom: 19. Okt., Seite 2663-2677 (DE-627)130994197 (DE-600)1082042-5 (DE-576)029154375 0925-1022 nnns volume:89 year:2021 number:12 day:19 month:10 pages:2663-2677 https://doi.org/10.1007/s10623-021-00938-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 89 2021 12 19 10 2663-2677 |
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walsh spectrum and nega spectrum of complementary arrays |
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Walsh spectrum and nega spectrum of complementary arrays |
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Abstract It has been shown that all the known binary Golay complementary sequences of length $$2^m$$ can be obtained by a single binary Golay complementary array of dimension m and size $$2\times 2 \times \cdots \times 2$$ which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length $$2^m$$ or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 |
| abstractGer |
Abstract It has been shown that all the known binary Golay complementary sequences of length $$2^m$$ can be obtained by a single binary Golay complementary array of dimension m and size $$2\times 2 \times \cdots \times 2$$ which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length $$2^m$$ or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 |
| abstract_unstemmed |
Abstract It has been shown that all the known binary Golay complementary sequences of length $$2^m$$ can be obtained by a single binary Golay complementary array of dimension m and size $$2\times 2 \times \cdots \times 2$$ which can be represented by a Boolean function. However, the construction of new binary Golay complementary sequences of length $$2^m$$ or Golay complementary arrays remains an open problem. In this paper, we studied the Walsh spectrum distribution and the nega spectrum distribution of the binary or quaternary Golay (Type-I) complementary array. Then, the Walsh spectrum of the binary Type-II complementary array and the nega spectrum of the binary Type-III complementary array are investigated as well. At last, the Walsh spectrum of a binary array in a complementary array set of size 4 is discussed. This work proves that binary and quaternary complementary arrays above-mentioned can only be constructed from (generalized) Boolean functions satisfying spectral values given in this paper. For instance, a binary Type-I complementary array must be bent for even m and near-bent for odd m with respect to the Walsh spectrum, and it must be negaplateaued, nega-bent or negalandscape with respect to the nega spectrum. On the other hand, constructions of new binary and quaternary complementary arrays may help us find new (generalized) Boolean functions with specific condition, such as bent or nega-bent functions. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 |
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Walsh spectrum and nega spectrum of complementary arrays |
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Wang, Zilong Xue, Erzhong |
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