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A MacWilliams-type identity for linear codes on weak order

Dae San Kim, Jeh Gwon Lee
2003 Discrete Mathematics  
Then we consider the P-weight enumerator for the linear code C of length n (n = n1 + · · · + nt) over Fq on P, and derive a MacWilliams-type identity relating the weight enumerator for the dual code C  ...  ⊥ of C on P and that for C on the dual poset P of P.  ...  MacWilliams-type identity for linear codes on the poset P = n 1 1 ⊕ · · · ⊕ n t 1.  ... 
doi:10.1016/s0012-365x(02)00498-3 fatcat:2lzopoqhqbfcdget5iongh4rv4

MacWilliams-type identities for fragment and sphere enumerators

Dae San Kim
2007 European journal of combinatorics (Print)  
We derive MacWilliams-type identities for the fragment and sphere enumerators, relating enumerators for the dual C ⊥ of the linear code C on P and those for C on the dual posetP.  ...  The linear changes of variables appearing in the identities are explicit.  ...  Voloch for helpful discussions and providing me with example 7.2. This work  ... 
doi:10.1016/j.ejc.2005.07.018 fatcat:4tp6dkgdf5gkrjuhxb3h5cnybe

Page 9830 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
New singly-even self-dual codes of lengths 102 to 110 are also given.” 2003m:94084 94B05 Kim, Dae San (KR-SGNG; Seoul) ; Lee, Jeh Gwon (KR-SGNG; Seoul) A MacWilliams-type identity for linear codes on weak  ...  In this paper, a generalization of MacWilliamsidentity for poset codes is obtained.  ... 

Author index to volume

2003 Discrete Mathematics  
Lee, A MacWilliams-type identity for linear codes on weak order (1-3) 181-194 Koukouvinos, C., see K. Betsumiya (1-3)  ...  ., Linear-time recognition of bipartite graphs plus two edges (1-3) 99-112 Dankelmann, P. and R.C. Laskar, Factor domination and minimum degree (1-3) 113-119 Du, B., see R.J.$ r! ak, T., I.  ... 
doi:10.1016/s0012-365x(02)00858-0 fatcat:2lp3aif63vdilb3yxukifhojse

Contents

2003 Discrete Mathematics  
Huang Maximum genus of a graph in terms of its embedding properties 171 D.S. Kim and J.G. Lee A MacWilliams-type identity for linear codes on weak order 181 M.  ...  Koukouvinos On self-dual codes over some prime fields 37 A. Brodsky, S. Durocher and E.  ... 
doi:10.1016/s0012-365x(02)00856-7 fatcat:sdrhosxi6bhpxkg534tmkzygei

Fourier-reflexive partitions and MacWilliams identities for additive codes

Heide Gluesing-Luerssen
2014 Designs, Codes and Cryptography  
Such partitions permit MacWilliams identities for the partition enumerators of additive codes.  ...  A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform.  ...  A major part of the final write-up took place during a research stay at the University of Zürich, and I am grateful to Joachim Rosenthal and his research group for the generous hospitality.  ... 
doi:10.1007/s10623-014-9940-x fatcat:xhd2hohd2nev5jme6mloj5rxjm

Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes [article]

Heide Gluesing-Luerssen
2013 arXiv   pre-print
Such partitions permit MacWilliams identities for the partition enumerators of additive codes.  ...  A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform.  ...  A major part of the final write-up took place during a research stay at the University of Zürich, and I am grateful to Joachim Rosenthal and his research group for the generous hospitality.  ... 
arXiv:1304.1207v1 fatcat:vi272ydcnjdydjzh3425zkt4ou

A MacWilliams Identity for Convolutional Codes: The General Case [article]

Heide Gluesing-Luerssen, Gert Schneider
2008 arXiv   pre-print
A MacWilliams Identity for convolutional codes will be established.  ...  The MacWilliams Identity applies to various notions of duality appearing in the literature on convolutional coding theory.  ...  In a previous article [8] we studied this invariant in detail, and, in particular, we could establish a weak MacWilliams type of identity for the generalized WAM.  ... 
arXiv:0805.3484v1 fatcat:d5zim6qxnrgs7c4o3defvmvb5i

On the MacWilliams Identity for Convolutional Codes

Heide Gluesing-Luerssen, Gert Schneider
2008 IEEE Transactions on Information Theory  
A MacWilliams Identity Conjecture, stating that the adjacency matrix of a code fully determines the adjacency matrix of the dual code, will be formulated, and an explicit formula for the transformation  ...  The conjecture will be proven for the class of convolutional codes where either the code itself or its dual does not have Forney indices bigger than one.  ...  This result can be regarded as a weak MacWilliams Identity for convolutional codes.  ... 
doi:10.1109/tit.2008.917664 fatcat:kqnuziml5jdp7p5mfrkubiwdge

On the MacWilliams Identity for Convolutional Codes [article]

Heide Gluesing-Luerssen, Gert Schneider
2006 arXiv   pre-print
A MacWilliams Identity Conjecture, stating that the adjacency matrix of a code fully determines the adjacency matrix of the dual code, will be formulated, and an explicit formula for the transformation  ...  The conjecture will be proven for the class of convolutional codes where either the code itself or its dual does not have Forney indices bigger than one.  ...  This result can be regarded as a weak MacWilliams Identity for convolutional codes.  ... 
arXiv:cs/0603013v1 fatcat:7dktjwmtzfaklmytzxje2xrwou

A Complete Linear Programming Hierarchy for Linear Codes [article]

Leonardo Nagami Coregliano and Fernando Granha Jeronimo and Chris Jones
2021 arXiv   pre-print
To date these results remain the best known lower and upper bounds on R_2(δ) with no improvement even for the important class of linear codes.  ...  We obtain our hierarchy by generalizing the Krawtchouk polynomials and MacWilliams inequalities to a suitable "higher-order" version taking into account interactions of ℓ words.  ...  Higher-order MacWilliams Identities and Inequalities In this section, we show a higher-order analogue of MacWilliams identities and inequalities using only basic Fourier analysis.  ... 
arXiv:2112.09221v1 fatcat:sqf3f22zn5d57k6ciuauioxp5a

Association schemes and MacWilliams dualities for generalized Niederreiter–Rosenbloom–Tsfasman posets

Dae San Kim, Hyun Kwang Kim
2012 Dissertationes Mathematicae  
(3.12) , (3.13) ) can now be invoked in order to get MacWilliams-type identities for linear codes in the association schemes X P , X H , X K , and X L , and thus, in particular, the proof for (1)⇒(2  ...  The MacWilliams identity for linear codes over F q is one of the most important identities in coding theory; it expresses the Hamming weight enumerator of the dual code C ⊥ of a linear code C over F q  ... 
doi:10.4064/dm487-0-1 fatcat:aj3x272bd5gztc4xj4upcbac4u

Page 5533 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
The code A, is identical in weight and distance distribution to the extended Preparata code.  ...  The quaternary code @ is the Z,4-analogue of the first-order Reed-Muller code. As a result, A, has a simple descrip- tion in the Z4-domain that admits a simple syndrome decoder.  ... 

Galois LCD codes over finite fields

Xiusheng Liu, Yun Fan, Hualu Liu
2018 Finite Fields and Their Applications  
We first introduce an upper bound on dimensions of divisible codes in terms of their weight spectrums, as well as a divisibility criteria for linear codes over arbitrary finite fields.  ...  This is generalized from weights modulo a prime power in linear codes, and yields much better results than the linear code version does.  ...  Note that there is a generalization of the MacWilliams identity for LeeC. Theorem 2.8 ( [Wan97] ). LetC be a Z 4 -linear code of length n.  ... 
doi:10.1016/j.ffa.2017.10.001 fatcat:wgn5lzdwpzgknpyl55w6yejc5a

The Art of Signaling: Fifty Years of Coding Theory [chapter]

2009 Information Theory  
The importance of these applications has served to focus the coding theory community on the complexity of coding techniques, for it is entirely appropriate that performance of a code should be valued as  ...  This framework did not encourage the creation of coding theory but it did not prevent telegraph code makers from using linear codes over a variety of moduli, and from realizing that the more paritycheck  ...  ACKNOWLEDGMENT The author is grateful to Peter Cameron for information on Fisher and the Hamming codes, to Jim Reeds for historical detail on telegraph codebooks, to Alexander Vardy for the statistics  ... 
doi:10.1109/9780470544907.ch20 fatcat:eeuyqk35orhmdleeiktamxgv6u
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