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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
Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal  ...  This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property.  ...  Bi-invariant Weights with Invertible Orthogonality Matrix on Principal Ideal Rings Let R be a finite principal ideal ring, and let w be a bi-invariant weight on R.  ... 
arXiv:1309.3292v1 fatcat:owdhd3k6rbdyhctbmvqjv2ryfa

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  
Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal  ...  This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property.  ...  Bi-invariant Weights with Invertible Orthogonality Matrix on Principal Ideal Rings Let R be a finite principal ideal ring, and let w be a bi-invariant weight on R .  ... 
doi:10.1016/j.jcta.2014.03.005 fatcat:jjxgvpmomnejhniulj43cafcqa

The Extension Theorem with Respect to Symmetrized Weight Compositions [chapter]

Noha ElGarem, Nefertiti Megahed, Jay A. Wood
2015 Coding Theory and Applications  
In the 1960s MacWilliams proved that finite fields have the extension property with respect to Hamming weight.  ...  The main theorem presented in this paper gives a sufficient condition for an alphabet to have the extension property with respect to symmetrized weight compositions.  ...  in the case of matrix rings over finite fields in [19] , and in the case of principal ideal rings, necessary and sufficient conditions were found for bi-invariant weights to satisfy the extension theorem  ... 
doi:10.1007/978-3-319-17296-5_18 dblp:conf/icmcta/ElGaremMW14 fatcat:dwmsz3wht5f7veb2uk3hwx72qm

The Extension Theorem for Bi-invariant Weights over Frobenius Rings and Frobenius Bimodules [article]

Oliver W. Gnilke, Marcus Greferath, Thomas Honold, Jay A. Wood, Jens Zumbrägel
2017 arXiv   pre-print
We give a sufficient condition for a bi-invariant weight on a Frobenius bimodule to satisfy the extension property.  ...  This condition applies to bi-invariant weights on a finite Frobenius ring as a special case.  ...  The conditions found generalize those for weights defined over a finite principal ideal ring in [4, Theorem 4.4] .  ... 
arXiv:1711.09939v1 fatcat:nl46wxrm3vgatbzbcw5o6qyw4i

Transform domain characterization of cyclic codes overZ m

B. Sundar Rajan, M. U. Siddiqi
1994 Applicable Algebra in Engineering, Communication and Computing  
It is shown that a cyclic code of length n over Zm, n relatively prime to m, consists of n-tuples over Z m having a specified set of DFT coefficients from the elements of an ideal of a subring of the extension  ...  Cyclic codes with symbols from a residue class integer ring Z m are characterized in terms of the discrete Fourier transform (DFT) of codewords defined over an appropriate extension ring of Z m.  ...  finite fields one can define BCH codes over Zpk as one whose all codewords have a specified set of spectral components taking values from the same specified ideal of the extension ring.  ... 
doi:10.1007/bf01225641 fatcat:wh4uu5klmbci3mbczvce6evuwq

Codes over $\mathbf {GF\pmb (4\pmb )}$ and $\mathbf {F}_2 \times \mathbf {F}_2$ and Hermitian lattices over imaginary quadratic fields

Kok Seng Chua
2004 Proceedings of the American Mathematical Society  
We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over GF(  ...  The results show that the two alphabets GF(4) and F 2 × F 2 are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.  ...  It also gives an isomorphism between the invariant ring of polynomials for the weight enumerators and the appropriate space of modular forms for the theta series of the constructed lattices.  ... 
doi:10.1090/s0002-9939-04-07724-x fatcat:ifsy3pwr3jcydludiuf5id3ttm

A theorem on the quantum evaluation of Weight Enumerators for a certain class of Cyclic Codes with a note on Cyclotomic cosets [article]

Joseph Geraci, Frank Van Bussel
2008 arXiv   pre-print
We provide a theorem on the quantum computation of the Weight Enumerator polynomial for a restricted family of cyclic codes.  ...  The complexity of obtaining an exact evaluation is O(k^2s( q)^2), where s is a parameter which determines the class of cyclic codes in question, q is the characteristic of the finite field over which the  ...  Thus, a principal ideal is generated by the one element a and a principal ideal ring is a ring in which every ideal is principal.  ... 
arXiv:cs/0703129v4 fatcat:zhwdtbbqrnhv3j3tw4cuszezlm

Metrics Based on Finite Directed Graphs and Coding Invariants [article]

Tuvi Etzion, Marcelo Firer, Roberto Assis Machado
2017 arXiv   pre-print
Finally, given a directed graph which determines a hierarchical poset, we present sufficient and necessary conditions to ensure the validity of the MacWilliams Identity and the MacWilliams Extension Property  ...  Given a finite directed graph with n vertices, we define a metric d_G on F_q^n, where F_q is the finite field with q elements.  ...  Some generalizations of poset metrics were introduced over the years, including its extension to Frobenius and finite principal ideal rings [4] , [9] and the poset-block metrics [1] .  ... 
arXiv:1609.08067v3 fatcat:smrev6xcd5fyva4cqzlkhubcui

New 5-designs

E.F. Assmus, H.F. Mattson
1969 Journal of Combinatorial Theory  
in the principal prime ideal.  ...  As such, they are principal ideals generated by the divisors of x ~ --1 over F.  ... 
doi:10.1016/s0021-9800(69)80115-8 fatcat:g22rvxqeyfcjdnvz5hebzzlbvq

On Quantum Stabilizer Codes derived from Local Frobenius Rings [article]

Heide Gluesing-Luerssen, Tefjol Pllaha
2017 arXiv   pre-print
We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field.  ...  In this paper we consider stabilizer codes over local Frobenius rings. First, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field.  ...  Theorem 2.1. Let R be a finite commutative ring. The following are equivalent. (a) socpRq -R{radpRq as R-modules. (b) socpRq is a principal ideal, i.e., socpRq " αR for some α P R.  ... 
arXiv:1710.09884v1 fatcat:oooh7cp6urfsdaxp7gzgqmvyim

Coding Theory

Joachim Rosenthal, Mohammad Amin Shokrollahi
2007 Oberwolfach Reports  
Minimal trellis realization by inspection for convolutional codes over finite rings Margreta Kuijper (joint work with Raquel Pinto) In this presentation I consider convolutional codes over finite rings  ...  Let ρ be a Type such that R is a direct product of matrix rings over chain rings (i.e. the left ideals are linearly ordered by inclusion).  ...  Here Σ denotes the set of channel output symbols; assume that this set either has finite cardinality, or is equal to R l or C l for some integer l ≥ 1.  ... 
doi:10.4171/owr/2007/56 fatcat:ac2rttsrn5g2rn5la72nudiyum

Eigenvalues of Cayley graphs [article]

Xiaogang Liu, Sanming Zhou
2019 arXiv   pre-print
A family of Cayley graphs on finite chain rings A finite chain ring is a finite local ring R such that for any ideals I, J of R we have either I ⊆ J or J ⊆ I.  ...  Let I = f (x)be the principal ideal generated by f (x). Then R[x]/I is a local ring of order |R| ns . Denote by Γ f (R) the unit group (R[x]/I) × .  ...  Theorem 11.6. ([24, Theorem 7]) Let Γ be a finite non-abelian group.  ... 
arXiv:1809.09829v2 fatcat:57hq3pcaurhybgfuzzh6qi74nu

Invariant semidefinite programs [article]

Christine Bachoc, Dion C. Gijswijt, Alexander Schrijver, Frank Vallentin
2010 arXiv   pre-print
In the last years many results in the area of semidefinite programming were obtained for invariant (finite dimensional, or infinite dimensional) semidefinite programs - SDPs which have symmetry.  ...  This was done for a variety of problems and applications.  ...  Acknowledgements We thank the referee for the helpful suggestions and comments.  ... 
arXiv:1007.2905v2 fatcat:nhbqxf5jqjgyrjfr3qnk4t4rie

Invariant Semidefinite Programs [chapter]

Christine Bachoc, Dion C. Gijswijt, Alexander Schrijver, Frank Vallentin
2011 International Series in Operations Research and Management Science  
This chapter provides the reader with the necessary background for dealing with semidefinite programs which have symmetry.  ...  Acknowledgements We thank the referee for the helpful suggestions and comments.  ...  The fourth author was supported by Vidi grant 639.032.917 from the Netherlands Organization for Scientific Research (NWO).  ... 
doi:10.1007/978-1-4614-0769-0_9 fatcat:vnun3jk5f5hwnjabutyxmtsesu

Index

1999 Linear Algebra and its Applications  
Sanchez-Giralda, Feedback invariants for linear dynamical systems over a principal ideal domain 218 (1995) 29 Hermida-Alonso, J.A., M.P. Perez and T.  ...  Liu, The extension of Roth's theorem for matrix equations over a ring 259 (1997) 229 Huang, L.J., see Ding, J. 212±213 (1994) 487 Huang, L.J., see Ding, J. 262 (1997) 229 Huang, T.  ...  ., On the nonexistence of rational ®rst integrals for systems of linear dierential equations 235 (1996) 107 Nummi, T., see Wang, S.-G. 289 (1999) 333 N uñez-Vald es, J., see G omez, J.  ... 
doi:10.1016/s0024-3795(99)00243-8 fatcat:xz5sbc3lzngydobg3ln6xp2cum
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