Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays
Let G be a finite abelian group and let exp (G) denote the least common multiple of the orders of all elements of G. A BH(G,h) matrix is a G-invariant | G| × | G| matrix H whose entries are complex hth roots of unity such that HH∗= | G| I|G|. By νp(x) we denote the p-adic valuation of the integer x....
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/143067 |
| _version_ | 1811682028301582336 |
|---|---|
| author | Duc, Tai Do |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Duc, Tai Do |
| author_sort | Duc, Tai Do |
| collection | NTU |
| description | Let G be a finite abelian group and let exp (G) denote the least common multiple of the orders of all elements of G. A BH(G,h) matrix is a G-invariant | G| × | G| matrix H whose entries are complex hth roots of unity such that HH∗= | G| I|G|. By νp(x) we denote the p-adic valuation of the integer x. Using bilinear forms over abelian groups, we [11] constructed new classes of BH(G,h) matrices under the following conditions.(i)νp(h) ≥ ⌈ νp(exp (G)) / 2 ⌉ for any prime divisor p of |G|, and(ii)ν2(h) ≥ 2 if ν2(| G|) is odd and G has a direct factor Z2. The purpose of this paper is to further study the conditions on G and h so that a BH(G,h) matrix exists. We will focus on BH(Zn,h) and BH(G,2pb) matrices, where p is an odd prime. Our results describe various relation among |G|, gcd (| G| , h) and lcm(|G|,h). Moreover, they confirm the nonexistence of 623 cases in the 3310 open cases for the existence of BH(Zn,h) matrices in which 1 ≤ n, h≤ 100. Finally, we show that BH(G,h) matrices can be used to construct a new family of perfect polyphase arrays. |
| first_indexed | 2024-10-01T03:50:19Z |
| format | Journal Article |
| id | ntu-10356/143067 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:50:19Z |
| publishDate | 2020 |
| record_format | dspace |
| spelling | ntu-10356/1430672023-02-28T19:51:03Z Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays Duc, Tai Do School of Physical and Mathematical Sciences Science::Mathematics Group Invariant Butson Hadamard Matrices Perfect Arrays Let G be a finite abelian group and let exp (G) denote the least common multiple of the orders of all elements of G. A BH(G,h) matrix is a G-invariant | G| × | G| matrix H whose entries are complex hth roots of unity such that HH∗= | G| I|G|. By νp(x) we denote the p-adic valuation of the integer x. Using bilinear forms over abelian groups, we [11] constructed new classes of BH(G,h) matrices under the following conditions.(i)νp(h) ≥ ⌈ νp(exp (G)) / 2 ⌉ for any prime divisor p of |G|, and(ii)ν2(h) ≥ 2 if ν2(| G|) is odd and G has a direct factor Z2. The purpose of this paper is to further study the conditions on G and h so that a BH(G,h) matrix exists. We will focus on BH(Zn,h) and BH(G,2pb) matrices, where p is an odd prime. Our results describe various relation among |G|, gcd (| G| , h) and lcm(|G|,h). Moreover, they confirm the nonexistence of 623 cases in the 3310 open cases for the existence of BH(Zn,h) matrices in which 1 ≤ n, h≤ 100. Finally, we show that BH(G,h) matrices can be used to construct a new family of perfect polyphase arrays. Accepted version 2020-07-28T01:00:13Z 2020-07-28T01:00:13Z 2020 Journal Article Duc, T. D. (2020). Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays. Designs, Codes, and Cryptography, 88, 73–90. doi:10.1007/s10623-019-00671-4 0925-1022 https://hdl.handle.net/10356/143067 10.1007/s10623-019-00671-4 2-s2.0-85071157249 88 73 90 en Designs, Codes, and Cryptography © 2019 Springer Science+Business Media. This is a post-peer-review, pre-copyedit version of an article published in Designs, Codes, and Cryptography. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10623-019-00671-4 application/pdf |
| spellingShingle | Science::Mathematics Group Invariant Butson Hadamard Matrices Perfect Arrays Duc, Tai Do Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays |
| title | Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays |
| title_full | Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays |
| title_fullStr | Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays |
| title_full_unstemmed | Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays |
| title_short | Necessary conditions for the existence of group-invariant Butson Hadamard matrices and a new family of perfect arrays |
| title_sort | necessary conditions for the existence of group invariant butson hadamard matrices and a new family of perfect arrays |
| topic | Science::Mathematics Group Invariant Butson Hadamard Matrices Perfect Arrays |
| url | https://hdl.handle.net/10356/143067 |
| work_keys_str_mv | AT ductaido necessaryconditionsfortheexistenceofgroupinvariantbutsonhadamardmatricesandanewfamilyofperfectarrays |