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Characteristics of Invariant Weights Related to Code Equivalence over Rings
[article]

2011
*
arXiv
*
pre-print

The

arXiv:1110.1538v1
fatcat:iweebpwj7rf6vnw7shirpiski4
*Equivalence*Theorem states that, for a given*weight*on the alphabet, every linear 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. ... Introduction Two linear*codes**of*the same length*over*a given alphabet are said*to*be*equivalent*if there exists a (*weight*preserving) monomial transformation mapping one*to*the other. ...##
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Characteristics of invariant weights related to code equivalence over rings

2012
*
Designs, Codes and Cryptography
*

The

doi:10.1007/s10623-012-9671-9
fatcat:ld6daw4jhncpzhuy46bgamte2a
*Equivalence*Theorem states that, for a given*weight*on the alphabet, every linear 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. ... Introduction Two linear*codes**of*the same length*over*a given alphabet are said*to*be*equivalent*if there exists a (*weight*preserving) monomial transformation mapping one*to*the other. ...##
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Codes over an infinite family of algebras

2017
*
Journal of Algebra Combinatorics Discrete Structures and Applications
*

Furthermore, we give generators for

doi:10.13069/jacodesmath.284947
fatcat:bfc65eeharajfbp56c55nywjz4
*invariant**ring**of*Euclidean self-dual*codes**over*B k through MacWilliams*relation**of*Hamming*weight*enumerator for such*codes*. 2010 MSC: 11T71 ... Using the previous connection, we give a characterization for*equivalence**of**codes**over*B k and Euclidean self-dual*codes*. ... We have*to*note that if C is an Euclidean self-dual*code*, then the*weight*enumerator*of*C is*invariant*under this transformation. ...##
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Jacobi forms over totally real fields and type II codes over Galois rings GR(2m,f)

2004
*
European journal of combinatorics (Print)
*

The

doi:10.1016/j.ejc.2003.01.001
fatcat:cmmbyhdwkfe6jijhyk6x4g2r6a
*invariant*polynomials can be constructed from the*weight*enumerators*of*type II*codes**over*the Galois*rings*. ... In this paper, we study the*invariant*polynomial*ring**of*the generalized Clifford-Weil group and give a connection*to*Jacobi modular forms*over*the totally real field. ... Acknowledgements We would like*to*thank Prof. P. Solé for his valuable comments. The first author was partially supported under the Com 2 MaC Postdoc Fund. ...##
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Self-dual Codes
[chapter]

2017
*
SpringerBriefs in Mathematics
*

C is

doi:10.1007/978-3-319-59806-2_5
fatcat:yur63snehbcsxfifkaqtwgbblq
*equivalent**to*the*code**over*F q with generator matrix [I n 2 I n 2 ], where I n 2 is the identity matrix*of*size n 2*over*F q . ... Let R be a local*ring*with*characteristic*congruent*to*3 (mod 4) then there exist self-dual*codes*for all lengths congruent*to*0 (mod 4)*over*R. ... For example, considering the*rings*Z 3 and Z 6 . The choice*of*Z 6 is a natural choice for the Euclidean*weight**of*Z 3 . ...##
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Codes over local rings of order 16 and binary codes

2016
*
Advances in Mathematics of Communications
*

We study

doi:10.3934/amc.2016012
fatcat:bejj5pk6drhqdc5afedcb7qzqu
*codes**over*the commutative local Frobenius*rings**of*order 16 with maximal ideals*of*size 8. We define a*weight*preserving Gray map and study the images*of*these*codes*as binary*codes*. ... We study self-dual*codes*and determine when they exist. 2010 Mathematics Subject Classification: Primary: 11T71, 94B05; Secondary: 13H99. ... Formally selfdual and self-dual*codes*have Hamming*weight*enumerators that are held*invariant*by the action*of*the MacWilliams*relations*. ...##
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Codes over Rk, Gray maps and their binary images

2011
*
Finite Fields and Their Applications
*

We introduce

doi:10.1016/j.ffa.2010.11.002
fatcat:qo5d4gqzk5akphvqxrulolxtla
*codes**over*an infinite family*of**rings*and describe two Gray maps*to*binary*codes*which are shown*to*be*equivalent*. ... The Lee*weights*for the elements*of*these*rings*are described and*related**to*the Hamming*weights**of*their binary image. ... Acknowledgments The authors would like*to*thank the anonymous referees for their valuable remarks. ...##
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On the apparent duality of the Kerdock and Preparata codes
[chapter]

1993
*
Lecture Notes in Computer Science
*

The construction involves quaternary

doi:10.1007/3-540-56686-4_30
fatcat:px5ghwocsfgkviqodgyhkkd67q
*codes**over*the*ring*Z Z4*of*integers modulo 4. We exhibit a quaternary*code*Q and its quaternary dual Q ? ... transform*of*that*of*the Kerdock*code*is only*to*be expected. ... Large sequence families for*code*-division multiple-access (CDMA) that are supersets*of*the near-optimum four-phase families described above, and which are*related**to*the Delsarte-Goethals*codes*are investigated ...##
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Translates of linear codes over Z/sub 4/

1997
*
IEEE Transactions on Information Theory
*

We give a method

doi:10.1109/18.605585
fatcat:qs2w4ubf4ffpfguvsimlz4k4q4
*to*compute the complete*weight*distribution*of*translates*of*linear*codes**over*Z Z Z4. ... the Preparata*codes*are both equal*to*ten, independent*of*the codelength. ... They also wish*to*thank the two anonymous referees for many constructive comments. ...##
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MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings
[article]

2013
*
arXiv
*
pre-print

A finite

arXiv:1309.3292v1
fatcat:owdhd3k6rbdyhctbmvqjv2ryfa
*ring*R and a*weight*w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear*codes*in R^n extends*to*a monomial transformation*of*R^n that preserves w. ... This paper addresses the question*of*a characterization*of*all bi-*invariant**weights*on a finite*ring*that satisfy the Extension Property. ... A*weight*w is called left (resp. right)*invariant*if G lt (w) = U (resp. G rt (w) = U ). A (left) linear*code**of*length n*over*R is a submodule C*of*R R n . ...##
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Codes and Siegel modular forms

1996
*
Discrete Mathematics
*

The (higher)

doi:10.1016/0012-365x(94)00271-j
fatcat:jrmswmvquzbybh2snqqflwbpuu
*weight*polynomials*of*such*codes*are proved*to*be the generators*of*the*ring**of**invariants**of*a polynomial*ring*in 2" variables under a certain specified finite group. ... It is proved that the*ring**of*Siegel modular forms in any genus is determined by doubly even self-dual*codes*and the theta*relations*. ... The equality*of*groups holds due*to*(WE) 3 = ~ and the*relations*in [34, Ch. 2]. For the*ring**of**invariants*we fix Go CPg = Bg . ...##
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On the algebraic structure of quasi-cyclic codes .I. Finite fields

2001
*
IEEE Transactions on Information Theory
*

A new algebraic approach

doi:10.1109/18.959257
fatcat:c6rhqs3xtzggxevrywfrwv2czm
*to*quasi-cyclic*codes*is introduced. The key idea is*to*regard a quasi-cyclic*code**over*a field as a linear*code**over*an auxiliary*ring*. ... That*ring*decomposition in turn yields a*code*construction from*codes**of*lower lengths which turns out*to*be in some cases the celebrated squaring and cubing constructions and in other cases the recent ... Ling wishes*to*thank CNRS-I3S, ESSI for hospitality. ...##
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MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

2014
*
Journal of combinatorial theory. Series A
*

A finite

doi:10.1016/j.jcta.2014.03.005
fatcat:jjxgvpmomnejhniulj43cafcqa
*ring*R and a*weight*w on R satisfy the Extension Property if every R -linear w -isometry between two R -linear*codes*in R n extends*to*a monomial transformation*of*R n that preserves w . ... This paper addresses the question*of*a characterization*of*all bi-*invariant**weights*on a finite*ring*that satisfy the Extension Property. ... A*weight*w is called left (resp. right)*invariant*if G lt (w) = U (resp. G rt (w) = U ). A (left) linear*code**of*length n*over*R is a submodule C*of*R R n . ...##
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Multi-twisted codes as free modules over principal ideal domains
[article]

2022
*
arXiv
*
pre-print

Thus, any

arXiv:2204.03156v1
fatcat:jsa3r2kmjzf6jklwjuod3bkdxm
*of*these*codes*corresponds*to*a free linear*code**over*the principal ideal domain (PID) 𝔽_q[x]. ... The Hermite normal form*of*matrices*over*PIDs is exploited*to*achieve the reduced GPMs*of*MT*codes*. Some properties*of*the reduced GPM are introduced, for example, the identical equation. ... The MacWilliams identity is a fundamental result in*coding*theory that*relates*the*weight*enumerator*of*a linear*code**to*the*weight*enumerator*of*its dual. ...##
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Page 5420 of Mathematical Reviews Vol. , Issue 2002G
[page]

2002
*
Mathematical Reviews
*

(RS-MOSC-NIT; Moscow); Nechaev, Aleksandr (RS-MOSC-NIT; Moscow) Complete

*weight*enumerators*of*generalized Kerdock*code*and*related*linear*codes**over*Galois*ring*. (English summary) Discrete Appl. ... The generalized Kerdock*codes*in the title are certain nonlin- ear*codes*that result from the concatenation*of*some linear*codes**over*the Galois*ring**of**characteristic*4 (within this context, any such ...
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