RETRACTED ARTICLE: Constructions for t-designs and s-resolvable t-designs
The purpose of the present paper is to introduce recursive methods for constructing simple t -designs, s -resolvable t -designs, and large sets of t -designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered desi...
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| Veröffentlicht in: | Designs, codes, and cryptography codes, and cryptography, 2024, Vol.92 (11), p.3503-3514 |
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| creator | Trung, Tran van |
| description | The purpose of the present paper is to introduce recursive methods for constructing simple
t
-designs,
s
-resolvable
t
-designs, and large sets of
t
-designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered designs. Consequently, many new infinite series of simple
t
-designs,
t
-designs with
s
-resolutions and large sets of
t
-designs can be derived from the new constructions. For example, by starting with an important result of Teirlinck stating that for every natural number
t
and for all
N
>
1
there is a large set
L
S
[
N
]
(
t
,
t
+
1
,
t
+
N
·
ℓ
(
t
)
)
, where
ℓ
(
t
)
=
∏
i
=
1
t
λ
(
i
)
·
λ
∗
(
i
)
,
λ
(
t
)
=
lcm
(
t
m
|
m
=
1
,
2
,
…
,
t
)
and
λ
∗
(
t
)
=
lcm
(
1
,
2
,
…
,
t
+
1
)
, we obtain the following statement. If
(
t
+
2
)
is composite, then there is a large set
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for all
N
>
1
. If
(
t
+
2
)
is prime, then there is an
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for any
N
with
gcd
(
t
+
2
,
N
)
=
1
. |
| doi_str_mv | 10.1007/s10623-024-01448-0 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_3111184009</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>3111184009</sourcerecordid><originalsourceid>FETCH-LOGICAL-c1150-35787aa0c398a943db37c12da9841ee973e68e5223616b9d477261cdce09ad253</originalsourceid><addsrcrecordid>eNp9kEFLAzEQhYMoWKt_wNOC5-hMkt0k3sq6aqEglPUc0iQtLXW3JlvBf290BW_OZWaY997AR8g1wi0CyLuEUDFOgQkKKISicEImWEpOZamqUzIBzUqKwNg5uUhpBwDIgU3IfNm0y1ndNg_FbNnO60VzX9R9l4Z4dMM2D8W6j8VAfUjbTd5s54tEY0j9_sOu9uHvdEnO1nafwtVvn5LXx6atn-ni5WlezxbUIZZAeSmVtBYc18pqwf2KS4fMW60EhqAlD5UKJWO8wmqlvZCSVei8C6CtZyWfkpsx9xD792NIg9n1x9jll4ZjLiUAdFaxUeVin1IMa3OI2zcbPw2C-UZmRmQmIzM_yAxkEx9NKYu7TYh_0f-4vgBy4muE</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>3111184009</pqid></control><display><type>article</type><title>RETRACTED ARTICLE: Constructions for t-designs and s-resolvable t-designs</title><source>SpringerLink Journals</source><creator>Trung, Tran van</creator><creatorcontrib>Trung, Tran van</creatorcontrib><description>The purpose of the present paper is to introduce recursive methods for constructing simple
t
-designs,
s
-resolvable
t
-designs, and large sets of
t
-designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered designs. Consequently, many new infinite series of simple
t
-designs,
t
-designs with
s
-resolutions and large sets of
t
-designs can be derived from the new constructions. For example, by starting with an important result of Teirlinck stating that for every natural number
t
and for all
N
>
1
there is a large set
L
S
[
N
]
(
t
,
t
+
1
,
t
+
N
·
ℓ
(
t
)
)
, where
ℓ
(
t
)
=
∏
i
=
1
t
λ
(
i
)
·
λ
∗
(
i
)
,
λ
(
t
)
=
lcm
(
t
m
|
m
=
1
,
2
,
…
,
t
)
and
λ
∗
(
t
)
=
lcm
(
1
,
2
,
…
,
t
+
1
)
, we obtain the following statement. If
(
t
+
2
)
is composite, then there is a large set
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for all
N
>
1
. If
(
t
+
2
)
is prime, then there is an
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for any
N
with
gcd
(
t
+
2
,
N
)
=
1
.</description><identifier>ISSN: 0925-1022</identifier><identifier>EISSN: 1573-7586</identifier><identifier>DOI: 10.1007/s10623-024-01448-0</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Coding and Information Theory ; Computer Science ; Cryptology ; Discrete Mathematics in Computer Science ; Infinite series ; Recursive methods</subject><ispartof>Designs, codes, and cryptography, 2024, Vol.92 (11), p.3503-3514</ispartof><rights>The Author(s) 2024</rights><rights>The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c1150-35787aa0c398a943db37c12da9841ee973e68e5223616b9d477261cdce09ad253</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10623-024-01448-0$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10623-024-01448-0$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Trung, Tran van</creatorcontrib><title>RETRACTED ARTICLE: Constructions for t-designs and s-resolvable t-designs</title><title>Designs, codes, and cryptography</title><addtitle>Des. Codes Cryptogr</addtitle><description>The purpose of the present paper is to introduce recursive methods for constructing simple
t
-designs,
s
-resolvable
t
-designs, and large sets of
t
-designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered designs. Consequently, many new infinite series of simple
t
-designs,
t
-designs with
s
-resolutions and large sets of
t
-designs can be derived from the new constructions. For example, by starting with an important result of Teirlinck stating that for every natural number
t
and for all
N
>
1
there is a large set
L
S
[
N
]
(
t
,
t
+
1
,
t
+
N
·
ℓ
(
t
)
)
, where
ℓ
(
t
)
=
∏
i
=
1
t
λ
(
i
)
·
λ
∗
(
i
)
,
λ
(
t
)
=
lcm
(
t
m
|
m
=
1
,
2
,
…
,
t
)
and
λ
∗
(
t
)
=
lcm
(
1
,
2
,
…
,
t
+
1
)
, we obtain the following statement. If
(
t
+
2
)
is composite, then there is a large set
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for all
N
>
1
. If
(
t
+
2
)
is prime, then there is an
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for any
N
with
gcd
(
t
+
2
,
N
)
=
1
.</description><subject>Coding and Information Theory</subject><subject>Computer Science</subject><subject>Cryptology</subject><subject>Discrete Mathematics in Computer Science</subject><subject>Infinite series</subject><subject>Recursive methods</subject><issn>0925-1022</issn><issn>1573-7586</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kEFLAzEQhYMoWKt_wNOC5-hMkt0k3sq6aqEglPUc0iQtLXW3JlvBf290BW_OZWaY997AR8g1wi0CyLuEUDFOgQkKKISicEImWEpOZamqUzIBzUqKwNg5uUhpBwDIgU3IfNm0y1ndNg_FbNnO60VzX9R9l4Z4dMM2D8W6j8VAfUjbTd5s54tEY0j9_sOu9uHvdEnO1nafwtVvn5LXx6atn-ni5WlezxbUIZZAeSmVtBYc18pqwf2KS4fMW60EhqAlD5UKJWO8wmqlvZCSVei8C6CtZyWfkpsx9xD792NIg9n1x9jll4ZjLiUAdFaxUeVin1IMa3OI2zcbPw2C-UZmRmQmIzM_yAxkEx9NKYu7TYh_0f-4vgBy4muE</recordid><startdate>2024</startdate><enddate>2024</enddate><creator>Trung, Tran van</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2024</creationdate><title>RETRACTED ARTICLE: Constructions for t-designs and s-resolvable t-designs</title><author>Trung, Tran van</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c1150-35787aa0c398a943db37c12da9841ee973e68e5223616b9d477261cdce09ad253</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Coding and Information Theory</topic><topic>Computer Science</topic><topic>Cryptology</topic><topic>Discrete Mathematics in Computer Science</topic><topic>Infinite series</topic><topic>Recursive methods</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Trung, Tran van</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Designs, codes, and cryptography</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Trung, Tran van</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>RETRACTED ARTICLE: Constructions for t-designs and s-resolvable t-designs</atitle><jtitle>Designs, codes, and cryptography</jtitle><stitle>Des. Codes Cryptogr</stitle><date>2024</date><risdate>2024</risdate><volume>92</volume><issue>11</issue><spage>3503</spage><epage>3514</epage><pages>3503-3514</pages><issn>0925-1022</issn><eissn>1573-7586</eissn><abstract>The purpose of the present paper is to introduce recursive methods for constructing simple
t
-designs,
s
-resolvable
t
-designs, and large sets of
t
-designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered designs. Consequently, many new infinite series of simple
t
-designs,
t
-designs with
s
-resolutions and large sets of
t
-designs can be derived from the new constructions. For example, by starting with an important result of Teirlinck stating that for every natural number
t
and for all
N
>
1
there is a large set
L
S
[
N
]
(
t
,
t
+
1
,
t
+
N
·
ℓ
(
t
)
)
, where
ℓ
(
t
)
=
∏
i
=
1
t
λ
(
i
)
·
λ
∗
(
i
)
,
λ
(
t
)
=
lcm
(
t
m
|
m
=
1
,
2
,
…
,
t
)
and
λ
∗
(
t
)
=
lcm
(
1
,
2
,
…
,
t
+
1
)
, we obtain the following statement. If
(
t
+
2
)
is composite, then there is a large set
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for all
N
>
1
. If
(
t
+
2
)
is prime, then there is an
L
S
[
N
]
(
t
,
t
+
2
,
t
+
1
+
N
·
ℓ
(
t
)
)
for any
N
with
gcd
(
t
+
2
,
N
)
=
1
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10623-024-01448-0</doi><tpages>12</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0925-1022 |
| ispartof | Designs, codes, and cryptography, 2024, Vol.92 (11), p.3503-3514 |
| issn | 0925-1022 1573-7586 |
| language | eng |
| recordid | cdi_proquest_journals_3111184009 |
| source | SpringerLink Journals |
| subjects | Coding and Information Theory Computer Science Cryptology Discrete Mathematics in Computer Science Infinite series Recursive methods |
| title | RETRACTED ARTICLE: Constructions for t-designs and s-resolvable t-designs |
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