Characteristics of invariant weights related to code equivalence over rings

The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hammin...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Designs, codes, and cryptography Jg. 66; H. 1-3; S. 145 - 156
Hauptverfasser: Greferath, Marcus, Mc Fadden, Cathy, Zumbrägel, Jens
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Boston Springer US 01.01.2013
Schlagworte:
Circuits
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Information and Communication
MacWilliams’ equivalence theorem
Weights
11T71
94B05
Ring-linear codes
Extension theorem
05E99
ISSN:0925-1022, 1573-7586
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Abstract The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006 ).
AbstractList The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006 ).
Author Greferath, Marcus
Mc Fadden, Cathy
Zumbrägel, Jens
Author_xml – sequence: 1
  givenname: Marcus
  surname: Greferath
  fullname: Greferath, Marcus
  organization: School of Mathematical Sciences, University College Dublin, Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography, and Information Security
– sequence: 2
  givenname: Cathy
  surname: Mc Fadden
  fullname: Mc Fadden, Cathy
  email: cathy.mcfadden@ucd.ie
  organization: School of Mathematical Sciences, University College Dublin, Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography, and Information Security
– sequence: 3
  givenname: Jens
  surname: Zumbrägel
  fullname: Zumbrägel, Jens
  organization: School of Mathematical Sciences, University College Dublin, Claude Shannon Institute for Discrete Mathematics, Coding, Cryptography, and Information Security
BookMark eNp9kM1OAyEURonRxLb6AO54gVEudJiZpWn8i03c6JowcGlpRqhAa3x7p6krF119q_Ml50zJeYgBCbkBdguMNXcZmOSiYsCrTjZQdWdkAnUjqqZu5TmZsI7XFTDOL8k05w1jDATjE_K6WOukTcHkc_Em0-ioD3udvA6FfqNfrUumCQdd0NISqYkWKX7t_F4PGAzSuMdEkw-rfEUunB4yXv_tjHw8Prwvnqvl29PL4n5ZGSGgVKaXrpe2FUwCaGsdM4jYuo5z3TkQonaWt6NPy4UGBvN-Li0Ttm_bnkvHxYw0x1-TYs4JnTK-6OJjKEn7QQFThybq2ESNTdShiepGEv6R2-Q_dfo5yfAjk7cHS0xqE3cpjIInoF_dzHba
CitedBy_id crossref_primary_10_1016_j_jcta_2014_03_005
crossref_primary_10_1016_j_jpaa_2014_04_026
crossref_primary_10_1142_S0219498818501980
crossref_primary_10_1007_s00200_020_00465_5
crossref_primary_10_1007_s10623_021_00945_w
crossref_primary_10_1109_TIT_2024_3507278
Cites_doi 10.1007/BF02992827
10.1006/jcta.1999.3033
10.1109/18.312154
10.1007/BF01229251
10.1353/ajm.1999.0024
10.1142/S0219498804000873
10.1002/j.1538-7305.1963.tb04003.x
10.1016/S0097-3165(96)80011-2
ContentType Journal Article
Copyright Springer Science+Business Media, LLC 2012
Copyright_xml – notice: Springer Science+Business Media, LLC 2012
DBID AAYXX
CITATION
DOI 10.1007/s10623-012-9671-9
DatabaseName CrossRef
DatabaseTitle CrossRef
DatabaseTitleList
DeliveryMethod fulltext_linktorsrc
Discipline Engineering
Mathematics
Computer Science
EISSN 1573-7586
EndPage 156
ExternalDocumentID 10_1007_s10623_012_9671_9
GroupedDBID -52
-5D
-5G
-BR
-EM
-Y2
-~C
.86
.DC
.VR
06D
0R~
0VY
199
1N0
1SB
2.D
203
28-
29F
2J2
2JN
2JY
2KG
2KM
2LR
2P1
2VQ
2~H
30V
4.4
406
408
409
40D
40E
5GY
5QI
5VS
67Z
6NX
78A
95-
95.
95~
96X
AAAVM
AABHQ
AACDK
AAHNG
AAIAL
AAJBT
AAJKR
AANZL
AARHV
AARTL
AASML
AATNV
AATVU
AAUYE
AAWCG
AAYIU
AAYQN
AAYTO
AAYZH
ABAKF
ABBBX
ABBXA
ABDZT
ABECU
ABFTD
ABFTV
ABHLI
ABHQN
ABJOX
ABKCH
ABKTR
ABMNI
ABMQK
ABNWP
ABQBU
ABQSL
ABSXP
ABTAH
ABTEG
ABTHY
ABTKH
ABTMW
ABULA
ABWNU
ABXPI
ACAOD
ACBXY
ACDTI
ACGFS
ACHSB
ACHXU
ACIWK
ACKNC
ACMDZ
ACMLO
ACOKC
ACOMO
ACPIV
ACSNA
ACZOJ
ADHHG
ADHIR
ADIMF
ADINQ
ADKNI
ADKPE
ADRFC
ADTPH
ADURQ
ADYFF
ADZKW
AEBTG
AEFIE
AEFQL
AEGAL
AEGNC
AEJHL
AEJRE
AEKMD
AEMSY
AENEX
AEOHA
AEPYU
AESKC
AETLH
AEVLU
AEXYK
AFEXP
AFGCZ
AFLOW
AFQWF
AFWTZ
AFZKB
AGAYW
AGDGC
AGGDS
AGJBK
AGMZJ
AGQEE
AGQMX
AGRTI
AGWIL
AGWZB
AGYKE
AHAVH
AHBYD
AHKAY
AHSBF
AHYZX
AIAKS
AIGIU
AIIXL
AILAN
AITGF
AJBLW
AJRNO
AJZVZ
ALMA_UNASSIGNED_HOLDINGS
ALWAN
AMKLP
AMXSW
AMYLF
AMYQR
AOCGG
ARMRJ
ASPBG
AVWKF
AXYYD
AYJHY
AZFZN
B-.
BA0
BAPOH
BBWZM
BDATZ
BGNMA
BSONS
CAG
COF
CS3
CSCUP
DDRTE
DL5
DNIVK
DPUIP
DU5
EBLON
EBS
EIOEI
EJD
ESBYG
FEDTE
FERAY
FFXSO
FIGPU
FINBP
FNLPD
FRRFC
FSGXE
FWDCC
GGCAI
GGRSB
GJIRD
GNWQR
GQ6
GQ7
GQ8
GXS
H13
HF~
HG5
HG6
HMJXF
HQYDN
HRMNR
HVGLF
HZ~
I09
IHE
IJ-
IKXTQ
ITM
IWAJR
IXC
IZIGR
IZQ
I~X
I~Z
J-C
J0Z
JBSCW
JCJTX
JZLTJ
KDC
KOV
KOW
LAK
LLZTM
M4Y
MA-
N2Q
N9A
NB0
NDZJH
NPVJJ
NQJWS
NU0
O9-
O93
O9G
O9I
O9J
OAM
OVD
P19
P2P
P9R
PF0
PT4
PT5
QOK
QOS
R4E
R89
R9I
RHV
RNI
RNS
ROL
RPX
RSV
RZC
RZE
RZK
S16
S1Z
S26
S27
S28
S3B
SAP
SCJ
SCLPG
SDD
SDH
SDM
SHX
SISQX
SJYHP
SMT
SNE
SNPRN
SNX
SOHCF
SOJ
SPISZ
SRMVM
SSLCW
STPWE
SZN
T13
T16
TEORI
TSG
TSK
TSV
TUC
U2A
UG4
UOJIU
UTJUX
UZXMN
VC2
VFIZW
W23
W48
WK8
YLTOR
Z45
Z7R
Z7U
Z7X
Z7Z
Z83
Z88
Z8M
Z8R
Z8T
Z8W
Z92
ZMTXR
ZWQNP
ZY4
~EX
AAPKM
AAYXX
ABBRH
ABDBE
ABFSG
ABJCF
ABRTQ
ACSTC
ADHKG
ADKFA
AEZWR
AFDZB
AFFHD
AFHIU
AFKRA
AFOHR
AGQPQ
AHPBZ
AHWEU
AIXLP
ARAPS
ATHPR
AYFIA
BENPR
BGLVJ
CCPQU
CITATION
HCIFZ
K7-
M7S
PHGZM
PHGZT
PQGLB
PTHSS
ID FETCH-LOGICAL-c331t-cb6fb6d830611addf0ceee8f922a9f1335fd28106823a1014b46d03db88b26f23
IEDL.DBID RSV
ISICitedReferencesCount 9
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000313056200011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0925-1022
IngestDate Tue Nov 18 22:21:31 EST 2025
Sat Nov 29 02:35:47 EST 2025
Fri Feb 21 02:34:12 EST 2025
IsDoiOpenAccess false
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 1-3
Keywords MacWilliams’ equivalence theorem
Weights
11T71
94B05
Ring-linear codes
Extension theorem
05E99
Language English
License http://www.springer.com/tdm
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c331t-cb6fb6d830611addf0ceee8f922a9f1335fd28106823a1014b46d03db88b26f23
OpenAccessLink https://inria.hal.science/inria-00607730
PageCount 12
ParticipantIDs crossref_citationtrail_10_1007_s10623_012_9671_9
crossref_primary_10_1007_s10623_012_9671_9
springer_journals_10_1007_s10623_012_9671_9
PublicationCentury 2000
PublicationDate 20130100
2013-1-00
PublicationDateYYYYMMDD 2013-01-01
PublicationDate_xml – month: 1
  year: 2013
  text: 20130100
PublicationDecade 2010
PublicationPlace Boston
PublicationPlace_xml – name: Boston
PublicationSubtitle An International Journal
PublicationTitle Designs, codes, and cryptography
PublicationTitleAbbrev Des. Codes Cryptogr
PublicationYear 2013
Publisher Springer US
Publisher_xml – name: Springer US
References Hammons, Kumar, Calderbank, Sloane, Solé (CR8) 1994; 40
Wood (CR17) 1999; 121
CR4
CR3
CR5
Stanley (CR14) 2012
Roman (CR13) 1992
CR18
CR16
McDonald (CR10) 1974
Ward, Wood (CR15) 1996; 73
Constantinescu, Heise (CR2) 1996; 57
Greferath, Schmidt (CR6) 2000; 92
Clark, Drake (CR1) 1973; 39
Greferath, Nechaev, Wisbauer (CR7) 2004; 3
Wood (CR19) 2008; 136
Nechaev (CR11) 1973; 91
MacWilliams (CR9) 1963; 42
Nechaev, Honold (CR12) 1999; 35
A.A. Nechaev (9671_CR12) 1999; 35
J. MacWilliams (9671_CR9) 1963; 42
I. Constantinescu (9671_CR2) 1996; 57
B.R. McDonald (9671_CR10) 1974
9671_CR3
9671_CR4
A.R. Hammons Jr. (9671_CR8) 1994; 40
9671_CR5
A.A. Nechaev (9671_CR11) 1973; 91
9671_CR18
9671_CR19
M. Greferath (9671_CR6) 2000; 92
9671_CR16
M. Greferath (9671_CR7) 2004; 3
H.N. Ward (9671_CR15) 1996; 73
R.P. Stanley (9671_CR14) 2012
S. Roman (9671_CR13) 1992
J.A. Wood (9671_CR17) 1999; 121
W.E. Clark (9671_CR1) 1973; 39
References_xml – volume: 42
  start-page: 79
  year: 1963
  end-page: 94
  ident: CR9
  article-title: A theorem on the distribution of weights in a systematic code
  publication-title: Bell Syst. Tech. J.
– ident: CR18
– volume: 39
  start-page: 147
  year: 1973
  end-page: 153
  ident: CR1
  article-title: Finite chain rings
  publication-title: Abh. Math. Sem. Univ. Hamburg,
  doi: 10.1007/BF02992827
– volume: 91
  start-page: 350
  issue: 133
  year: 1973
  end-page: 366, 471
  ident: CR11
  article-title: Finite rings of principal ideals
  publication-title: Mat. Sb. (N.S.)
– volume: 92
  start-page: 17
  issue: 1
  year: 2000
  end-page: 28
  ident: CR6
  article-title: Finite-ring combinatorics and MacWilliams’ equivalence theorem
  publication-title: J. Comb. Theory Ser. A
  doi: 10.1006/jcta.1999.3033
– ident: CR3
– ident: CR4
– ident: CR16
– volume: 40
  start-page: 301
  issue: 2
  year: 1994
  end-page: 319
  ident: CR8
  article-title: The -linearity of Kerdock, Preparata, Goethals, and related codes
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/18.312154
– volume: 57
  start-page: 63
  issue: 1–2
  year: 1996
  end-page: 69
  ident: CR2
  article-title: On the concept of code-isomorphy
  publication-title: J. Geom.
  doi: 10.1007/BF01229251
– volume: 35
  start-page: 18
  issue: 3
  year: 1999
  end-page: 39
  ident: CR12
  article-title: Fully weighted modules and representations of codes
  publication-title: Problemy Peredachi Informatsii
– volume: 73
  start-page: 348
  issue: 2
  year: 1996
  end-page: 352
  ident: CR15
  article-title: Characters and the equivalence of codes
  publication-title: J. Comb. Theory Ser. A
– volume: 121
  start-page: 555
  issue: 3
  year: 1999
  end-page: 575
  ident: CR17
  article-title: Duality for modules over finite rings and applications to coding theory
  publication-title: Am. J. Math.
  doi: 10.1353/ajm.1999.0024
– volume: 3
  start-page: 247
  issue: 3
  year: 2004
  end-page: 272
  ident: CR7
  article-title: Finite quasi-Frobenius modules and linear codes
  publication-title: J. Algebra Appl.
  doi: 10.1142/S0219498804000873
– year: 2012
  ident: CR14
  publication-title: Enumerative Combinatorics. Volume 1, Volume 9 of Cambridge Studies in Advanced Mathematics
– volume: 136
  start-page: 699
  issue: 2
  year: 2008
  end-page: 706
  ident: CR19
  article-title: Code equivalence characterizes finite Frobenius rings
  publication-title: Proc. Am. Math. Soc.
– ident: CR5
– year: 1992
  ident: CR13
  publication-title: Coding and Information Theory Volume 134 of Graduate Texts in Mathematics
– year: 1974
  ident: CR10
  publication-title: Finite Rings with Identity
– volume: 42
  start-page: 79
  year: 1963
  ident: 9671_CR9
  publication-title: Bell Syst. Tech. J.
  doi: 10.1002/j.1538-7305.1963.tb04003.x
– ident: 9671_CR16
– volume-title: Finite Rings with Identity
  year: 1974
  ident: 9671_CR10
– volume: 57
  start-page: 63
  issue: 1–2
  year: 1996
  ident: 9671_CR2
  publication-title: J. Geom.
  doi: 10.1007/BF01229251
– volume: 121
  start-page: 555
  issue: 3
  year: 1999
  ident: 9671_CR17
  publication-title: Am. J. Math.
  doi: 10.1353/ajm.1999.0024
– volume: 3
  start-page: 247
  issue: 3
  year: 2004
  ident: 9671_CR7
  publication-title: J. Algebra Appl.
  doi: 10.1142/S0219498804000873
– volume: 73
  start-page: 348
  issue: 2
  year: 1996
  ident: 9671_CR15
  publication-title: J. Comb. Theory Ser. A
  doi: 10.1016/S0097-3165(96)80011-2
– ident: 9671_CR19
– volume: 91
  start-page: 350
  issue: 133
  year: 1973
  ident: 9671_CR11
  publication-title: Mat. Sb. (N.S.)
– ident: 9671_CR18
– ident: 9671_CR5
– volume: 92
  start-page: 17
  issue: 1
  year: 2000
  ident: 9671_CR6
  publication-title: J. Comb. Theory Ser. A
  doi: 10.1006/jcta.1999.3033
– volume-title: Enumerative Combinatorics. Volume 1, Volume 9 of Cambridge Studies in Advanced Mathematics
  year: 2012
  ident: 9671_CR14
– ident: 9671_CR3
– ident: 9671_CR4
– volume: 35
  start-page: 18
  issue: 3
  year: 1999
  ident: 9671_CR12
  publication-title: Problemy Peredachi Informatsii
– volume: 40
  start-page: 301
  issue: 2
  year: 1994
  ident: 9671_CR8
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/18.312154
– volume-title: Coding and Information Theory Volume 134 of Graduate Texts in Mathematics
  year: 1992
  ident: 9671_CR13
– volume: 39
  start-page: 147
  year: 1973
  ident: 9671_CR1
  publication-title: Abh. Math. Sem. Univ. Hamburg,
  doi: 10.1007/BF02992827
SSID ssj0001302
Score 1.9905173
Snippet The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire...
SourceID crossref
springer
SourceType Enrichment Source
Index Database
Publisher
StartPage 145
SubjectTerms Circuits
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Information and Communication
Title Characteristics of invariant weights related to code equivalence over rings
URI https://link.springer.com/article/10.1007/s10623-012-9671-9
Volume 66
WOSCitedRecordID wos000313056200011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAVX
  databaseName: SpringerLINK Contemporary 1997-Present
  customDbUrl:
  eissn: 1573-7586
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0001302
  issn: 0925-1022
  databaseCode: RSV
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22
  providerName: Springer Nature
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3dS8MwED90-qAPfkzF-UUefFICbdKmyaMMhyAO8Yu9laZJYCCdrt38902ydm6ggj4Wrkd7vY_eXe53AOexJIYkPMOC8ABHOmNY0oxiI31lP2FKSb9sIun3-WAg7us57rI57d60JL2nXhh2s6Hapr4EC5aEWKzCmo123Fnjw-PL3P26TpwH2CMOYpOQppX5HYvlYLTcCfUBprf9r0fbga36fxJdzRRgF1Z00YbtZlcDqk23DZsLwIP26m6O1lruwW13GbUZjQwaFlObRVuxow9fPC2Rn3rRClUj5ObgkX6fDK2aOvbInQNFjnW5D8-966fuDa6XLOCc0rDCuWRGMsVt6hCG1tmZwIZNzY0gJBPGZrCxUYTbF-SEZm6xr4yYCqiSnEvCDKEH0CpGhT4EZHOzKGNhxnRCIxkLkYhcmUQrHsVZREwHgkbaaV4jkLtFGK_pF3ayE2RqBZk6QaaiAxfzW95m8Bu_EV82nyetLbH8mfroT9THsEH8IgxXfDmBVjWe6FNYz6fVsByfeQ38BOpI1Lo
linkProvider Springer Nature
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3dS8MwED90CuqD06k4P_PgkxJY0zZNHmU4JvtAdMreStMkMJBN123--yZZOzdQQR8L16O9Xu56X78DuAoF0SRiCeaE1XCgEoqFn_hYC5fZj6iUwi2biLpd1u_zh3yOOyu63YuSpLPUS8NuxlWb0JdgTiMP83XYCIzDsn18j08vC_NrK3EOYI9YiE1CilLmdyxWndFqJdQ5mEb5X4-2B7v5_yS6nSvAPqypYQXKxa4GlB_dCuwsAQ-aq84CrTU7gFZ9FbUZjTQaDGcmijZiRx8ueZohN_WiJJqMkJ2DR-p9OjBqatkj2weKLOvsEJ4bd716E-dLFnDq-94Ep4JqQSUzoYPnGWOna8ZtKqY5IQnXJoINtSTMvCAjfmIX-4qAypovBWOCUE38IygNR0N1DMjEZkFCvYSqyA9EyHnEU6kjJVkQJgHRVagV0o7THIHcLsJ4jb-wk60gYyPI2Aoy5lW4XtzyNoff-I34pvg8cX4Ss5-pT_5EfQlbzV6nHbfvu61T2CZuKYZNxJxBaTKeqnPYTGeTQTa-cNr4Ceyf154
linkToPdf http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3dS8MwED90iuiD06k4P_PgkxLWpl2aPMp0KNMx8IO9laZJYCDdXLv575t07dxABfGxcD3SyyXXy-V-P4CLpiCaBCzCnDAH-yqiWHiRh7XIT_YDKqXIySaCbpf1-7xX8Jym5W33siQ562mwKE1J1hhJ3VhofDNh26TBBHMauJivwppvOYNsuv70Ot-KbVUuB9sjFm6TkLKs-Z2K5cC0XBXNg027-u9h7sB28Z-JrmeOsQsrKqlBteRwQMWSrsHWAiCheXqco7ime9BpLaM5o6FGg2RqsmszHegjP1RNUd4NoyTKhsj2xyP1PhkY97Xqkb0fiqzqdB9e2rfPrTtckC_g2PPcDMeCakElMymF65pNUDsmnCqmOSER1yazbWpJmPlARrzIEv4Kn0rHk4IxQagm3gFUkmGiDgGZnM2PqBtRFXi-aHIe8FjqQEnmNyOf6Do4peXDuEAmtwQZb-EXprI1ZGgMGVpDhrwOl_NXRjNYjt-Er8qpCosVmv4sffQn6XPY6N20w4f7bucYNknOlWHPZ06gko0n6hTW42k2SMdnuWN-AnjR4II
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Characteristics+of+invariant+weights+related+to+code+equivalence+over+rings&rft.jtitle=Designs%2C+codes%2C+and+cryptography&rft.au=Greferath%2C+Marcus&rft.au=Mc+Fadden%2C+Cathy&rft.au=Zumbr%C3%A4gel%2C+Jens&rft.date=2013-01-01&rft.pub=Springer+US&rft.issn=0925-1022&rft.eissn=1573-7586&rft.volume=66&rft.issue=1-3&rft.spage=145&rft.epage=156&rft_id=info:doi/10.1007%2Fs10623-012-9671-9&rft.externalDocID=10_1007_s10623_012_9671_9
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0925-1022&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0925-1022&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0925-1022&client=summon