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The Construction of Regular Hadamard Matrices by Cyclotomic Classes
For every prime power q ≡ 7 m o d 16 , there are ( q ; a , b , c , d )-partitions of GF ( q ), with odd integers a , b , c , and d , where a ≡ ± 1 m o d 8 such that q = a 2 + 2 ( b 2 + c 2 + d 2 ) and d 2 = b 2 + 2 a c + 2 b d . Many results for the existence of 4 - { q 2 ; q ( q - 1 ) 2 ; q ( q -...
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| Published in: | Bulletin of the Iranian Mathematical Society 2021-06, Vol.47 (3), p.601-625 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| cited_by | cdi_FETCH-LOGICAL-c330t-2c57f664b723fceea02da6d94505d472716e7f392cbb16b057a6ce22500c283a3 |
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| cites | cdi_FETCH-LOGICAL-c330t-2c57f664b723fceea02da6d94505d472716e7f392cbb16b057a6ce22500c283a3 |
| container_end_page | 625 |
| container_issue | 3 |
| container_start_page | 601 |
| container_title | Bulletin of the Iranian Mathematical Society |
| container_volume | 47 |
| creator | Xia, Tianbing Xia, Mingyuan Seberry, Jennifer |
| description | For every prime power
q
≡
7
m
o
d
16
, there are (
q
;
a
,
b
,
c
,
d
)-partitions of
GF
(
q
), with odd integers
a
,
b
,
c
, and
d
, where
a
≡
±
1
m
o
d
8
such that
q
=
a
2
+
2
(
b
2
+
c
2
+
d
2
)
and
d
2
=
b
2
+
2
a
c
+
2
b
d
. Many results for the existence of
4
-
{
q
2
;
q
(
q
-
1
)
2
;
q
(
q
-
2
)
}
SDS which are simple homogeneous polynomials of parameters
a
,
b
,
c
and
d
of degree at most 2 have been found. Hence, for each value of
q
, the construction of SDS becomes equivalent to building a
(
q
;
a
,
b
,
c
,
d
)
-partition. Once this is done, the verification of the construction only involves verifying simple conditions on
a
,
b
,
c
and
d
which can be done manually. |
| doi_str_mv | 10.1007/s41980-020-00402-9 |
| format | article |
| fullrecord | <record><control><sourceid>crossref_sprin</sourceid><recordid>TN_cdi_crossref_primary_10_1007_s41980_020_00402_9</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>10_1007_s41980_020_00402_9</sourcerecordid><originalsourceid>FETCH-LOGICAL-c330t-2c57f664b723fceea02da6d94505d472716e7f392cbb16b057a6ce22500c283a3</originalsourceid><addsrcrecordid>eNp9kM1KxDAURoMoOIzzAq7yAtGb_3YpRR1hRJAR3IUkTcdKp5GkXfTtjY5rF5d7F9-5fByErincUAB9mwWtKyDAyoAARuoztKKaS1JJKs_LDVQTUPB-iTY59w6EYLSqhFihZv8RcBPHPKXZT30ccezwazjMg014a1t7tKnFz3ZKvQ8ZuwU3ix_iFI-9x81gcw75Cl10dshh87fX6O3hft9sye7l8am52xHPOUyEeak7pYTTjHc-BAustaqthQTZCs00VUF3vGbeOaocSG2VD4xJAM8qbvkasdNfn2LOKXTmK_Wl32IomB8T5mTCFBPm14SpC8RPUC7h8RCS-YxzGkvP_6hvkAVgow</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>The Construction of Regular Hadamard Matrices by Cyclotomic Classes</title><source>Springer Nature</source><creator>Xia, Tianbing ; Xia, Mingyuan ; Seberry, Jennifer</creator><creatorcontrib>Xia, Tianbing ; Xia, Mingyuan ; Seberry, Jennifer</creatorcontrib><description>For every prime power
q
≡
7
m
o
d
16
, there are (
q
;
a
,
b
,
c
,
d
)-partitions of
GF
(
q
), with odd integers
a
,
b
,
c
, and
d
, where
a
≡
±
1
m
o
d
8
such that
q
=
a
2
+
2
(
b
2
+
c
2
+
d
2
)
and
d
2
=
b
2
+
2
a
c
+
2
b
d
. Many results for the existence of
4
-
{
q
2
;
q
(
q
-
1
)
2
;
q
(
q
-
2
)
}
SDS which are simple homogeneous polynomials of parameters
a
,
b
,
c
and
d
of degree at most 2 have been found. Hence, for each value of
q
, the construction of SDS becomes equivalent to building a
(
q
;
a
,
b
,
c
,
d
)
-partition. Once this is done, the verification of the construction only involves verifying simple conditions on
a
,
b
,
c
and
d
which can be done manually.</description><identifier>ISSN: 1017-060X</identifier><identifier>EISSN: 1735-8515</identifier><identifier>DOI: 10.1007/s41980-020-00402-9</identifier><language>eng</language><publisher>Singapore: Springer Singapore</publisher><subject>Mathematics ; Mathematics and Statistics ; Original Paper</subject><ispartof>Bulletin of the Iranian Mathematical Society, 2021-06, Vol.47 (3), p.601-625</ispartof><rights>Iranian Mathematical Society 2020</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c330t-2c57f664b723fceea02da6d94505d472716e7f392cbb16b057a6ce22500c283a3</citedby><cites>FETCH-LOGICAL-c330t-2c57f664b723fceea02da6d94505d472716e7f392cbb16b057a6ce22500c283a3</cites><orcidid>0000-0002-9558-4293 ; 0000-0002-4520-5021</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Xia, Tianbing</creatorcontrib><creatorcontrib>Xia, Mingyuan</creatorcontrib><creatorcontrib>Seberry, Jennifer</creatorcontrib><title>The Construction of Regular Hadamard Matrices by Cyclotomic Classes</title><title>Bulletin of the Iranian Mathematical Society</title><addtitle>Bull. Iran. Math. Soc</addtitle><description>For every prime power
q
≡
7
m
o
d
16
, there are (
q
;
a
,
b
,
c
,
d
)-partitions of
GF
(
q
), with odd integers
a
,
b
,
c
, and
d
, where
a
≡
±
1
m
o
d
8
such that
q
=
a
2
+
2
(
b
2
+
c
2
+
d
2
)
and
d
2
=
b
2
+
2
a
c
+
2
b
d
. Many results for the existence of
4
-
{
q
2
;
q
(
q
-
1
)
2
;
q
(
q
-
2
)
}
SDS which are simple homogeneous polynomials of parameters
a
,
b
,
c
and
d
of degree at most 2 have been found. Hence, for each value of
q
, the construction of SDS becomes equivalent to building a
(
q
;
a
,
b
,
c
,
d
)
-partition. Once this is done, the verification of the construction only involves verifying simple conditions on
a
,
b
,
c
and
d
which can be done manually.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><issn>1017-060X</issn><issn>1735-8515</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kM1KxDAURoMoOIzzAq7yAtGb_3YpRR1hRJAR3IUkTcdKp5GkXfTtjY5rF5d7F9-5fByErincUAB9mwWtKyDAyoAARuoztKKaS1JJKs_LDVQTUPB-iTY59w6EYLSqhFihZv8RcBPHPKXZT30ccezwazjMg014a1t7tKnFz3ZKvQ8ZuwU3ix_iFI-9x81gcw75Cl10dshh87fX6O3hft9sye7l8am52xHPOUyEeak7pYTTjHc-BAustaqthQTZCs00VUF3vGbeOaocSG2VD4xJAM8qbvkasdNfn2LOKXTmK_Wl32IomB8T5mTCFBPm14SpC8RPUC7h8RCS-YxzGkvP_6hvkAVgow</recordid><startdate>20210601</startdate><enddate>20210601</enddate><creator>Xia, Tianbing</creator><creator>Xia, Mingyuan</creator><creator>Seberry, Jennifer</creator><general>Springer Singapore</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-9558-4293</orcidid><orcidid>https://orcid.org/0000-0002-4520-5021</orcidid></search><sort><creationdate>20210601</creationdate><title>The Construction of Regular Hadamard Matrices by Cyclotomic Classes</title><author>Xia, Tianbing ; Xia, Mingyuan ; Seberry, Jennifer</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c330t-2c57f664b723fceea02da6d94505d472716e7f392cbb16b057a6ce22500c283a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Xia, Tianbing</creatorcontrib><creatorcontrib>Xia, Mingyuan</creatorcontrib><creatorcontrib>Seberry, Jennifer</creatorcontrib><collection>CrossRef</collection><jtitle>Bulletin of the Iranian Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Xia, Tianbing</au><au>Xia, Mingyuan</au><au>Seberry, Jennifer</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Construction of Regular Hadamard Matrices by Cyclotomic Classes</atitle><jtitle>Bulletin of the Iranian Mathematical Society</jtitle><stitle>Bull. Iran. Math. Soc</stitle><date>2021-06-01</date><risdate>2021</risdate><volume>47</volume><issue>3</issue><spage>601</spage><epage>625</epage><pages>601-625</pages><issn>1017-060X</issn><eissn>1735-8515</eissn><abstract>For every prime power
q
≡
7
m
o
d
16
, there are (
q
;
a
,
b
,
c
,
d
)-partitions of
GF
(
q
), with odd integers
a
,
b
,
c
, and
d
, where
a
≡
±
1
m
o
d
8
such that
q
=
a
2
+
2
(
b
2
+
c
2
+
d
2
)
and
d
2
=
b
2
+
2
a
c
+
2
b
d
. Many results for the existence of
4
-
{
q
2
;
q
(
q
-
1
)
2
;
q
(
q
-
2
)
}
SDS which are simple homogeneous polynomials of parameters
a
,
b
,
c
and
d
of degree at most 2 have been found. Hence, for each value of
q
, the construction of SDS becomes equivalent to building a
(
q
;
a
,
b
,
c
,
d
)
-partition. Once this is done, the verification of the construction only involves verifying simple conditions on
a
,
b
,
c
and
d
which can be done manually.</abstract><cop>Singapore</cop><pub>Springer Singapore</pub><doi>10.1007/s41980-020-00402-9</doi><tpages>25</tpages><orcidid>https://orcid.org/0000-0002-9558-4293</orcidid><orcidid>https://orcid.org/0000-0002-4520-5021</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1017-060X |
| ispartof | Bulletin of the Iranian Mathematical Society, 2021-06, Vol.47 (3), p.601-625 |
| issn | 1017-060X 1735-8515 |
| language | eng |
| recordid | cdi_crossref_primary_10_1007_s41980_020_00402_9 |
| source | Springer Nature |
| subjects | Mathematics Mathematics and Statistics Original Paper |
| title | The Construction of Regular Hadamard Matrices by Cyclotomic Classes |
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