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The MacWilliams Theorem for Four-Dimensional Modulo Metrics [article]

Murat Guzeltepe, Mehmet Ozen
2014 arXiv   pre-print
In this paper, the MacWilliams theorem is stated for codes over finite field with four-dimensional modulo metrics.  ...  In [4] , the MacWilliam theorem proved for codes over finite fields with two-dimensional modulo metric.  ...  Conclusion In this paper, we proved the MacWilliams for four-dimensional modulo metrics. In fact, the quaternion Mannheim metric can be seen as a four-dimensional generalization of the Lee metric.  ... 
arXiv:0912.5009v2 fatcat:eufuatjgd5byxjgmqna5a74y64

Page 6070 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
This paper deals with the MacWilliams theorem for codes over finite fields with two-dimensional modulo metrics.  ...  By a variation on the exponential sums method and combinatorial arguments, we sharpen known bounds on these parameters.” 98i:94037 94B10 Huber, Klaus The MacWilliams theorem for two-dimensional modulo  ... 

On dual codes in the Doob schemes [article]

Denis S. Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)
2019 arXiv   pre-print
In particular, for each case, we determine the dual scheme, on the same set but with different metric, such that the weight distribution of an additive code C in the Doob scheme D(m,n'+n") is related by  ...  the MacWilliams identities with the weight distribution of the dual code C^ in the dual scheme.  ...  INTRODUCTION The codes in Doob graphs are special cases of codes over Eisenstein-Jacobi integers, see, e.g., [4] , [7] , which can be used for the information transmission in the channels with two-dimensional  ... 
arXiv:1902.00020v1 fatcat:w6l7ugj37ng73lsdl4ymfg73eq

Rank-metric codes and their duality theory [article]

Alberto Ravagnani
2015 arXiv   pre-print
We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes.  ...  We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter.  ...  Acknowledgement The author is grateful to Elisa Gorla and to the Referees for many useful suggestions that improved the presentation of the paper.  ... 
arXiv:1410.1333v3 fatcat:ayh7uoneh5afnntl42ebj7vv6i

Page 81 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
For a two-dimensional lattice L,,(a,b), given by all integer multi- ples of a vector (a,b)' modulo m, 0 < a,b < m, the author studies the problem of finding a shortest nonzero vector (with respect to the  ...  It is shown that the problem can be reduced to the computation of a Minkowski-reduced basis of a closely related planar lattice, and, therefore, it can be solved by a well-known algorithm for two-dimensional  ... 

Galois LCD codes over finite fields

Xiusheng Liu, Yun Fan, Hualu Liu
2018 Finite Fields and Their Applications  
For higher level codes we prove an induction theorem by using the binary version of the divisibility criteria, as well as Ward's bound and the new generated bound.  ...  Moreover, this induction theorem allows us to determine the exact bound and the codes that attain the bound for level three codes of relatively small length. vi Contents Acknowledgements iv Abstract v  ...  We see that the bound can be attained when m = 2n, by letting C be the concatenation of two n-folded replicated (e + 1)-dimensional dual Hamming codes.  ... 
doi:10.1016/j.ffa.2017.10.001 fatcat:wgn5lzdwpzgknpyl55w6yejc5a

Page 3114 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews  
The ideas underlying the proof of the theorem yield necessary condi- tions on whether a solution to the MacWilliams identity can be the weight distribution of a linear code.  ...  We intro- duce a two-dimensional modular distance called the Mannheim distance and propose using codes designed for this distance.  ... 

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

2009 Information Theory  
The addition of adjacent transpositions to Hamming errors means that the universe of words makes for a more complicated metric space, so that determining efficiency or even optimality of a particular code  ...  The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero.  ...  The author takes full responsibility for the faults that remain.  ... 
doi:10.1109/9780470544907.ch20 fatcat:eeuyqk35orhmdleeiktamxgv6u

MacWilliams Type Identities for Some New $m$-Spotty Weight Enumerators

Anuradha Sharma, Amit K. Sharma
2012 IEEE Transactions on Information Theory  
We also discuss some of their applications and establish MacWilliams identities for each of the above-mentioned enumerators.  ...  ) for byte error-control codes over the finite chain ring F q + uF q + u 2 F q + · · · + u e−1 F q (u e = 0) or the ring F q +uF q +vF q +uvF q (u 2 = 0, v 2 = 0, uv = vu), where F q is the finite field  ...  Acknowledgement We would like to thank the reviewers for their useful comments and suggestions, which have improved the presentation of the paper.  ... 
doi:10.1109/tit.2012.2189924 fatcat:d3y533jchfazna7f6evnq4xjxu

Cyclic Codes over Some Finite Rings [article]

Murat Guzeltepe, Mehmet Ozen
2009 arXiv   pre-print
In this paper cyclic codes are established with respect to the Mannheim metric over some finite rings by using Gaussian integers and the decoding algorithm for these codes is given.  ...  In 1997, Huber proved the MacWilliams theorem for two-dimensional modulo metric (Mannheim metric) [3] .  ...  In Proposition 4, the algebraic background, which is essential for obtaining cyclic code over the other finite rings, is arranged and in Theorem 3, it is shown how to obtain cyclic codes over the other  ... 
arXiv:0905.4163v1 fatcat:ggq4kpdqvrfd5euwyelyck2kf4

Traces of Hecke operators and refined weight enumerators of Reed-Solomon codes

Nathan Kaplan, Ian Petrow
2017 Transactions of the American Mathematical Society  
The main tool we use is the Eichler-Selberg trace formula, which gives along the way a variation of a theorem of Birch on the distribution of rational point counts for elliptic curves with prescribed 2  ...  Our main results are formulas for the coefficients of the the quadratic residue weight enumerators for such codes.  ...  The authors also thank him for carefully reading a draft of this paper and the anonymous referee for helpful comments.  ... 
doi:10.1090/tran/7089 fatcat:rgou5vtuxjhb3buggvn5mm2yca

The art of signaling: fifty years of coding theory

A.R. Calderbank
1998 IEEE Transactions on Information Theory  
The addition of adjacent transpositions to Hamming errors means that the universe of words makes for a more complicated metric space, so that determining efficiency or even optimality of a particular code  ...  The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero.  ...  The author takes full responsibility for the faults that remain.  ... 
doi:10.1109/18.720549 fatcat:utsr3jz3frexjezv3bznqkaxie

Translates of linear codes over Z/sub 4/

A. Bonnecaze, I.M. Duursma
1997 IEEE Transactions on Information Theory  
For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for  ...  The method follows known ideas that have already been used successfully by others for Hamming weight distributions.  ...  They also wish to thank the two anonymous referees for many constructive comments.  ... 
doi:10.1109/18.605585 fatcat:qs2w4ubf4ffpfguvsimlz4k4q4

Weight distributions of linear codes and the Gleason-Pierce theorem

George T Kennedy
1994 Journal of combinatorial theory. Series A  
The Gleason-Pierce theorem characterizes those fields for which formally selfdual divisible codes can exist.  ...  The ideas underlying the proof of the theorem yield necessary conditions on whether a solution to the MacWilliams identity can be the weight distribution of a linear code.  ...  ACKNOWLEDGMENTS I thank the University of Illinois at Chicago for their hospitality while this work was in progress.  ... 
doi:10.1016/0097-3165(94)90004-3 fatcat:irh3kyfkdjdqxpm6pzk37vj5xi

Generalized Pair Weights of Linear Codes and Linear Isomorphisms Preserving Pair Weights [article]

Hongwei Liu, Xu Pan
2020 arXiv   pre-print
Finally, a necessary and sufficient condition for a linear isomorphism preserving pair weights between two linear codes is obtained.  ...  In this paper, we first introduce the notion of generalized pair weights of an [n, k]-linear code over the finite field 𝔽_q and the notion of pair r-equiweight codes, where 1≤ r≤ k-1.  ...  Beelen, "A note on the generalized Hamming weights of Reed-Muller codes," Applicable Algebra in Engineering, Communication and Computing, vol. 30, no. 3, pp. 233-242, 2019.  ... 
arXiv:2002.02165v2 fatcat:qulwmqb3ljatplcltoqeuakedy
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