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

Marcus Greferath, Cathy Mc Fadden, Jens Zumbrägel
2011 arXiv   pre-print
The 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.  ... 
arXiv:1110.1538v1 fatcat:iweebpwj7rf6vnw7shirpiski4

Characteristics of invariant weights related to code equivalence over rings

Marcus Greferath, Cathy Mc Fadden, Jens Zumbrägel
2012 Designs, Codes and Cryptography  
The 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.  ... 
doi:10.1007/s10623-012-9671-9 fatcat:ld6daw4jhncpzhuy46bgamte2a

Codes over an infinite family of algebras

Irwansyah, Intan Muchtadi-Alamsyah, Ahmad Muchlis, Aleams Barra, Djoko Suprijanto
2017 Journal of Algebra Combinatorics Discrete Structures and Applications  
Furthermore, we give generators for 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.  ... 
doi:10.13069/jacodesmath.284947 fatcat:bfc65eeharajfbp56c55nywjz4

Jacobi forms over totally real fields and type II codes over Galois rings GR(2m,f)

K. Betsumiya, Y. Choie
2004 European journal of combinatorics (Print)  
The 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.  ... 
doi:10.1016/j.ejc.2003.01.001 fatcat:cmmbyhdwkfe6jijhyk6x4g2r6a

Self-dual Codes [chapter]

Steven T. Dougherty
2017 SpringerBriefs in Mathematics  
C is 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 .  ... 
doi:10.1007/978-3-319-59806-2_5 fatcat:yur63snehbcsxfifkaqtwgbblq

Codes over local rings of order 16 and binary codes

Steven T. Dougherty, Esengül Saltürk, Steve Szabo
2016 Advances in Mathematics of Communications  
We study 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.  ... 
doi:10.3934/amc.2016012 fatcat:bejj5pk6drhqdc5afedcb7qzqu

Codes over Rk, Gray maps and their binary images

Steven T. Dougherty, Bahattin Yildiz, Suat Karadeniz
2011 Finite Fields and Their Applications  
We introduce 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.  ... 
doi:10.1016/j.ffa.2010.11.002 fatcat:qo5d4gqzk5akphvqxrulolxtla

On the apparent duality of the Kerdock and Preparata codes [chapter]

A. Roger Hammons, P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, Patrick Solé
1993 Lecture Notes in Computer Science  
The construction involves quaternary 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  ... 
doi:10.1007/3-540-56686-4_30 fatcat:px5ghwocsfgkviqodgyhkkd67q

Translates of linear codes over Z/sub 4/

A. Bonnecaze, I.M. Duursma
1997 IEEE Transactions on Information Theory  
We give a method 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.  ... 
doi:10.1109/18.605585 fatcat:qs2w4ubf4ffpfguvsimlz4k4q4

MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings [article]

Marcus Greferath, Thomas Honold, Cathy Mc Fadden, Jay A. Wood, and Jens Zumbrägel
2013 arXiv   pre-print
A finite 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 .  ... 
arXiv:1309.3292v1 fatcat:owdhd3k6rbdyhctbmvqjv2ryfa

Codes and Siegel modular forms

Bernhard Runge
1996 Discrete Mathematics  
The (higher) 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 .  ... 
doi:10.1016/0012-365x(94)00271-j fatcat:jrmswmvquzbybh2snqqflwbpuu

On the algebraic structure of quasi-cyclic codes .I. Finite fields

San Ling, P. Sole
2001 IEEE Transactions on Information Theory  
A new algebraic approach 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.  ... 
doi:10.1109/18.959257 fatcat:c6rhqs3xtzggxevrywfrwv2czm

MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

Marcus Greferath, Thomas Honold, Cathy Mc Fadden, Jay A. Wood, Jens Zumbrägel
2014 Journal of combinatorial theory. Series A  
A finite 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 .  ... 
doi:10.1016/j.jcta.2014.03.005 fatcat:jjxgvpmomnejhniulj43cafcqa

Multi-twisted codes as free modules over principal ideal domains [article]

Ramy Taki ElDin
2022 arXiv   pre-print
Thus, any 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.  ... 
arXiv:2204.03156v1 fatcat:jsa3r2kmjzf6jklwjuod3bkdxm

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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