Several Goethals–Seidel Sequences with Special Structures
In this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted to designing...
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MDPI AG
2024-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/4/530 |
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| author | Shuhui Shen Xiaojun Zhang |
| author_facet | Shuhui Shen Xiaojun Zhang |
| author_sort | Shuhui Shen |
| collection | DOAJ |
| description | In this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted to designing some sequences that can potentially construct all GS sequences. Firstly, it is proven that a quad of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>±</mo><mn>1</mn></mrow></semantics></math></inline-formula> polynomials can be considered a linear combination of eight polynomials with coefficients uniquely belonging to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mo>±</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>. Based on this fact, we change the construction of a quad of Goethals–Seidel sequences to find eight sequences consisting of 0 and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>±</mo><mn>1</mn></mrow></semantics></math></inline-formula>. One more motivation is to obtain these sequences more efficiently. To this end, we make use of the <i>k</i>-block, of which some properties of (anti) symmetry are discussed. After this, we can then look for the sequences with the help of computers since the symmetry properties facilitate reducing the search range. Moreover, we find that one of the eight blocks, which we utilize to construct GS sequences directly, can also be combined with Williamson sequences to generate GS sequences with more order. Several examples are provided to verify the theoretical results. The main contribution of this work is in building a bridge linking the GS sequences and eight polynomials, and the paper also provides a novel insight through which to consider the existence of GS sequences. |
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| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-07T22:22:53Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-82a98d8c19304bac998865b59e1b170d2024-02-23T15:26:04ZengMDPI AGMathematics2227-73902024-02-0112453010.3390/math12040530Several Goethals–Seidel Sequences with Special StructuresShuhui Shen0Xiaojun Zhang1School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, ChinaSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, ChinaIn this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted to designing some sequences that can potentially construct all GS sequences. Firstly, it is proven that a quad of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>±</mo><mn>1</mn></mrow></semantics></math></inline-formula> polynomials can be considered a linear combination of eight polynomials with coefficients uniquely belonging to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mo>±</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>. Based on this fact, we change the construction of a quad of Goethals–Seidel sequences to find eight sequences consisting of 0 and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>±</mo><mn>1</mn></mrow></semantics></math></inline-formula>. One more motivation is to obtain these sequences more efficiently. To this end, we make use of the <i>k</i>-block, of which some properties of (anti) symmetry are discussed. After this, we can then look for the sequences with the help of computers since the symmetry properties facilitate reducing the search range. Moreover, we find that one of the eight blocks, which we utilize to construct GS sequences directly, can also be combined with Williamson sequences to generate GS sequences with more order. Several examples are provided to verify the theoretical results. The main contribution of this work is in building a bridge linking the GS sequences and eight polynomials, and the paper also provides a novel insight through which to consider the existence of GS sequences.https://www.mdpi.com/2227-7390/12/4/530Goethals–Seidel sequencesk-block and k-partitionsymmetry and antisymmetry |
| spellingShingle | Shuhui Shen Xiaojun Zhang Several Goethals–Seidel Sequences with Special Structures Mathematics Goethals–Seidel sequences k-block and k-partition symmetry and antisymmetry |
| title | Several Goethals–Seidel Sequences with Special Structures |
| title_full | Several Goethals–Seidel Sequences with Special Structures |
| title_fullStr | Several Goethals–Seidel Sequences with Special Structures |
| title_full_unstemmed | Several Goethals–Seidel Sequences with Special Structures |
| title_short | Several Goethals–Seidel Sequences with Special Structures |
| title_sort | several goethals seidel sequences with special structures |
| topic | Goethals–Seidel sequences k-block and k-partition symmetry and antisymmetry |
| url | https://www.mdpi.com/2227-7390/12/4/530 |
| work_keys_str_mv | AT shuhuishen severalgoethalsseidelsequenceswithspecialstructures AT xiaojunzhang severalgoethalsseidelsequenceswithspecialstructures |