On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes
In this paper, we first discuss linear codes over R and present the decomposition structure of linear codes over the mixed alphabet F q R , where R = F q + u F q + v F q + u v F q , with u 2 = 1, v 2 = 1, u v = v u and q = p m for odd prime p , positive integer m . Let θ be an automorphism on F q ....
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| Veröffentlicht in: | Cryptography and communications 2023, Vol.15 (1), p.171-198 |
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| container_title | Cryptography and communications |
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| creator | Dinh, Hai Q. Bag, Tushar Abdukhalikov, Kanat Pathak, Sachin Upadhyay, Ashish K. Bandi, Ramakrishna Chinnakum, Warattaya |
| description | In this paper, we first discuss linear codes over
R
and present the decomposition structure of linear codes over the mixed alphabet
F
q
R
, where
R
=
F
q
+
u
F
q
+
v
F
q
+
u
v
F
q
, with
u
2
= 1,
v
2
= 1,
u
v
=
v
u
and
q
=
p
m
for odd prime
p
, positive integer
m
. Let
θ
be an automorphism on
F
q
. Extending
θ
to Θ over
R
, we study skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
, where
λ
and Γ are units in
F
q
and
R
, respectively. We also show that, the dual of a skew (
θ
,Θ)-(
λ
,Γ)-constacyclic code over
F
q
R
is a skew (
θ
,Θ)-(
λ
− 1
,Γ
− 1
)-constacyclic code over
F
q
R
. We classify some self-dual skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes using the possible values of units of
R
. Also using suitable values of
λ
,
θ
,Γ and Θ, we present the structure of other linear codes over
F
q
R
. We construct a Gray map over
F
q
R
and study the Gray images of skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew
θ
-cyclic codes than from cyclic codes over
F
q
. |
| doi_str_mv | 10.1007/s12095-022-00594-3 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2764473220</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2764473220</sourcerecordid><originalsourceid>FETCH-LOGICAL-c249t-c4c15faf246157781e4a5cafb721c60d7cd25e6e55deaedbd9217c231a3a9b053</originalsourceid><addsrcrecordid>eNp9kMlOwzAURS0EEqXwA6wssTZ4jMkSVUxSpW5gbb04TklJndR2GP4e0yDYsbKffM991kHonNFLRqm-iozTUhHKOaFUlZKIAzRjpSgIl0od_t6lPkYnMW4oLRSXYobGlceAbQcx4r7B8dW9Y9v7mMB-2q61eahdfnpzAW_bD1dj6IYXqFyKGHyehiGnILWZwa2f2DDa1Po17ofUbqHbB3cj-DRup75TdNRAF93ZzzlHz3e3T4sHslzdPy5ulsRyWSZipWWqgYbLgimtr5mToCw0lebMFrTWtubKFU6p2oGrq7rkTFsuGAgoK6rEHF1MvUPod6OLyWz6Mfi80nBdZBuCc5pTfErZ0McYXGOGkP8dPg2j5luvmfSarNfs9RqRITFBMYf92oW_6n-oL06-f6o</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2764473220</pqid></control><display><type>article</type><title>On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes</title><source>SpringerLink</source><creator>Dinh, Hai Q. ; Bag, Tushar ; Abdukhalikov, Kanat ; Pathak, Sachin ; Upadhyay, Ashish K. ; Bandi, Ramakrishna ; Chinnakum, Warattaya</creator><creatorcontrib>Dinh, Hai Q. ; Bag, Tushar ; Abdukhalikov, Kanat ; Pathak, Sachin ; Upadhyay, Ashish K. ; Bandi, Ramakrishna ; Chinnakum, Warattaya</creatorcontrib><description>In this paper, we first discuss linear codes over
R
and present the decomposition structure of linear codes over the mixed alphabet
F
q
R
, where
R
=
F
q
+
u
F
q
+
v
F
q
+
u
v
F
q
, with
u
2
= 1,
v
2
= 1,
u
v
=
v
u
and
q
=
p
m
for odd prime
p
, positive integer
m
. Let
θ
be an automorphism on
F
q
. Extending
θ
to Θ over
R
, we study skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
, where
λ
and Γ are units in
F
q
and
R
, respectively. We also show that, the dual of a skew (
θ
,Θ)-(
λ
,Γ)-constacyclic code over
F
q
R
is a skew (
θ
,Θ)-(
λ
− 1
,Γ
− 1
)-constacyclic code over
F
q
R
. We classify some self-dual skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes using the possible values of units of
R
. Also using suitable values of
λ
,
θ
,Γ and Θ, we present the structure of other linear codes over
F
q
R
. We construct a Gray map over
F
q
R
and study the Gray images of skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew
θ
-cyclic codes than from cyclic codes over
F
q
.</description><identifier>ISSN: 1936-2447</identifier><identifier>EISSN: 1936-2455</identifier><identifier>DOI: 10.1007/s12095-022-00594-3</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Automorphisms ; Binary system ; Circuits ; Codes ; Coding and Information Theory ; Communications Engineering ; Computer Science ; Data Structures and Information Theory ; Error correcting codes ; Error correction ; Information and Communication ; Linear codes ; Mathematics of Computing ; Networks</subject><ispartof>Cryptography and communications, 2023, Vol.15 (1), p.171-198</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022</rights><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c249t-c4c15faf246157781e4a5cafb721c60d7cd25e6e55deaedbd9217c231a3a9b053</citedby><cites>FETCH-LOGICAL-c249t-c4c15faf246157781e4a5cafb721c60d7cd25e6e55deaedbd9217c231a3a9b053</cites><orcidid>0000-0002-7613-8351</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12095-022-00594-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12095-022-00594-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Dinh, Hai Q.</creatorcontrib><creatorcontrib>Bag, Tushar</creatorcontrib><creatorcontrib>Abdukhalikov, Kanat</creatorcontrib><creatorcontrib>Pathak, Sachin</creatorcontrib><creatorcontrib>Upadhyay, Ashish K.</creatorcontrib><creatorcontrib>Bandi, Ramakrishna</creatorcontrib><creatorcontrib>Chinnakum, Warattaya</creatorcontrib><title>On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes</title><title>Cryptography and communications</title><addtitle>Cryptogr. Commun</addtitle><description>In this paper, we first discuss linear codes over
R
and present the decomposition structure of linear codes over the mixed alphabet
F
q
R
, where
R
=
F
q
+
u
F
q
+
v
F
q
+
u
v
F
q
, with
u
2
= 1,
v
2
= 1,
u
v
=
v
u
and
q
=
p
m
for odd prime
p
, positive integer
m
. Let
θ
be an automorphism on
F
q
. Extending
θ
to Θ over
R
, we study skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
, where
λ
and Γ are units in
F
q
and
R
, respectively. We also show that, the dual of a skew (
θ
,Θ)-(
λ
,Γ)-constacyclic code over
F
q
R
is a skew (
θ
,Θ)-(
λ
− 1
,Γ
− 1
)-constacyclic code over
F
q
R
. We classify some self-dual skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes using the possible values of units of
R
. Also using suitable values of
λ
,
θ
,Γ and Θ, we present the structure of other linear codes over
F
q
R
. We construct a Gray map over
F
q
R
and study the Gray images of skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew
θ
-cyclic codes than from cyclic codes over
F
q
.</description><subject>Automorphisms</subject><subject>Binary system</subject><subject>Circuits</subject><subject>Codes</subject><subject>Coding and Information Theory</subject><subject>Communications Engineering</subject><subject>Computer Science</subject><subject>Data Structures and Information Theory</subject><subject>Error correcting codes</subject><subject>Error correction</subject><subject>Information and Communication</subject><subject>Linear codes</subject><subject>Mathematics of Computing</subject><subject>Networks</subject><issn>1936-2447</issn><issn>1936-2455</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp9kMlOwzAURS0EEqXwA6wssTZ4jMkSVUxSpW5gbb04TklJndR2GP4e0yDYsbKffM991kHonNFLRqm-iozTUhHKOaFUlZKIAzRjpSgIl0od_t6lPkYnMW4oLRSXYobGlceAbQcx4r7B8dW9Y9v7mMB-2q61eahdfnpzAW_bD1dj6IYXqFyKGHyehiGnILWZwa2f2DDa1Po17ofUbqHbB3cj-DRup75TdNRAF93ZzzlHz3e3T4sHslzdPy5ulsRyWSZipWWqgYbLgimtr5mToCw0lebMFrTWtubKFU6p2oGrq7rkTFsuGAgoK6rEHF1MvUPod6OLyWz6Mfi80nBdZBuCc5pTfErZ0McYXGOGkP8dPg2j5luvmfSarNfs9RqRITFBMYf92oW_6n-oL06-f6o</recordid><startdate>2023</startdate><enddate>2023</enddate><creator>Dinh, Hai Q.</creator><creator>Bag, Tushar</creator><creator>Abdukhalikov, Kanat</creator><creator>Pathak, Sachin</creator><creator>Upadhyay, Ashish K.</creator><creator>Bandi, Ramakrishna</creator><creator>Chinnakum, Warattaya</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-7613-8351</orcidid></search><sort><creationdate>2023</creationdate><title>On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes</title><author>Dinh, Hai Q. ; Bag, Tushar ; Abdukhalikov, Kanat ; Pathak, Sachin ; Upadhyay, Ashish K. ; Bandi, Ramakrishna ; Chinnakum, Warattaya</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c249t-c4c15faf246157781e4a5cafb721c60d7cd25e6e55deaedbd9217c231a3a9b053</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Automorphisms</topic><topic>Binary system</topic><topic>Circuits</topic><topic>Codes</topic><topic>Coding and Information Theory</topic><topic>Communications Engineering</topic><topic>Computer Science</topic><topic>Data Structures and Information Theory</topic><topic>Error correcting codes</topic><topic>Error correction</topic><topic>Information and Communication</topic><topic>Linear codes</topic><topic>Mathematics of Computing</topic><topic>Networks</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Dinh, Hai Q.</creatorcontrib><creatorcontrib>Bag, Tushar</creatorcontrib><creatorcontrib>Abdukhalikov, Kanat</creatorcontrib><creatorcontrib>Pathak, Sachin</creatorcontrib><creatorcontrib>Upadhyay, Ashish K.</creatorcontrib><creatorcontrib>Bandi, Ramakrishna</creatorcontrib><creatorcontrib>Chinnakum, Warattaya</creatorcontrib><collection>CrossRef</collection><jtitle>Cryptography and communications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dinh, Hai Q.</au><au>Bag, Tushar</au><au>Abdukhalikov, Kanat</au><au>Pathak, Sachin</au><au>Upadhyay, Ashish K.</au><au>Bandi, Ramakrishna</au><au>Chinnakum, Warattaya</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes</atitle><jtitle>Cryptography and communications</jtitle><stitle>Cryptogr. Commun</stitle><date>2023</date><risdate>2023</risdate><volume>15</volume><issue>1</issue><spage>171</spage><epage>198</epage><pages>171-198</pages><issn>1936-2447</issn><eissn>1936-2455</eissn><abstract>In this paper, we first discuss linear codes over
R
and present the decomposition structure of linear codes over the mixed alphabet
F
q
R
, where
R
=
F
q
+
u
F
q
+
v
F
q
+
u
v
F
q
, with
u
2
= 1,
v
2
= 1,
u
v
=
v
u
and
q
=
p
m
for odd prime
p
, positive integer
m
. Let
θ
be an automorphism on
F
q
. Extending
θ
to Θ over
R
, we study skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
, where
λ
and Γ are units in
F
q
and
R
, respectively. We also show that, the dual of a skew (
θ
,Θ)-(
λ
,Γ)-constacyclic code over
F
q
R
is a skew (
θ
,Θ)-(
λ
− 1
,Γ
− 1
)-constacyclic code over
F
q
R
. We classify some self-dual skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes using the possible values of units of
R
. Also using suitable values of
λ
,
θ
,Γ and Θ, we present the structure of other linear codes over
F
q
R
. We construct a Gray map over
F
q
R
and study the Gray images of skew (
θ
,Θ)-(
λ
,Γ)-constacyclic codes over
F
q
R
. As applications of our study, we construct many good codes, among them, there are 17 optimal codes and 2 near-optimal codes. Finally, we discuss the advantages in a construction of quantum error-correcting codes (QECCs) from skew
θ
-cyclic codes than from cyclic codes over
F
q
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s12095-022-00594-3</doi><tpages>28</tpages><orcidid>https://orcid.org/0000-0002-7613-8351</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1936-2447 |
| ispartof | Cryptography and communications, 2023, Vol.15 (1), p.171-198 |
| issn | 1936-2447 1936-2455 |
| language | eng |
| recordid | cdi_proquest_journals_2764473220 |
| source | SpringerLink |
| subjects | Automorphisms Binary system Circuits Codes Coding and Information Theory Communications Engineering Computer Science Data Structures and Information Theory Error correcting codes Error correction Information and Communication Linear codes Mathematics of Computing Networks |
| title | On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes |
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