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

2013
*
arXiv
*
pre-print

Having solved this question in previous papers

arXiv:1309.3292v1
fatcat:owdhd3k6rbdyhctbmvqjv2ryfa
*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. ...##
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MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

2014
*
Journal of combinatorial theory. Series A
*

Having solved this question in previous papers

doi:10.1016/j.jcta.2014.03.005
fatcat:jjxgvpmomnejhniulj43cafcqa
*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 . ...##
###
The Extension Theorem with Respect to Symmetrized Weight Compositions
[chapter]

2015
*
Coding Theory and Applications
*

In the 1960s

doi:10.1007/978-3-319-17296-5_18
dblp:conf/icmcta/ElGaremMW14
fatcat:dwmsz3wht5f7veb2uk3hwx72qm
*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*...##
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The Extension Theorem for Bi-invariant Weights over Frobenius Rings and Frobenius Bimodules
[article]

2017
*
arXiv
*
pre-print

We give a sufficient condition

arXiv:1711.09939v1
fatcat:nl46wxrm3vgatbzbcw5o6qyw4i
*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] . ...##
###
Transform domain characterization of cyclic codes overZ m

1994
*
Applicable Algebra in Engineering, Communication and Computing
*

It is shown that a cyclic code of length n

doi:10.1007/bf01225641
fatcat:wh4uu5klmbci3mbczvce6evuwq
*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*. ...##
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Codes over $\mathbf {GF\pmb (4\pmb )}$ and $\mathbf {F}_2 \times \mathbf {F}_2$ and Hermitian lattices over imaginary quadratic fields

2004
*
Proceedings of the American Mathematical Society
*

We introduce a family of

doi:10.1090/s0002-9939-04-07724-x
fatcat:ifsy3pwr3jcydludiuf5id3ttm
*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. ...##
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A theorem on the quantum evaluation of Weight Enumerators for a certain class of Cyclic Codes with a note on Cyclotomic cosets
[article]

2008
*
arXiv
*
pre-print

We provide a

arXiv:cs/0703129v4
fatcat:zhwdtbbqrnhv3j3tw4cuszezlm
*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*. ...##
###
Metrics Based on Finite Directed Graphs and Coding Invariants
[article]

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

arXiv:1609.08067v3
fatcat:smrev6xcd5fyva4cqzlkhubcui
*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] . ...##
###
New 5-designs

1969
*
Journal of Combinatorial Theory
*

in the

doi:10.1016/s0021-9800(69)80115-8
fatcat:g22rvxqeyfcjdnvz5hebzzlbvq
*principal*prime*ideal*. ... As such, they are*principal**ideals*generated by the divisors of x ~ --1*over*F. ...##
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On Quantum Stabilizer Codes derived from Local Frobenius Rings
[article]

2017
*
arXiv
*
pre-print

We show that

arXiv:1710.09884v1
fatcat:oooh7cp6urfsdaxp7gzgqmvyim
*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. ...##
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Coding Theory

2007
*
Oberwolfach Reports
*

Minimal trellis realization by inspection

doi:10.4171/owr/2007/56
fatcat:ac2rttsrn5g2rn5la72nudiyum
*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. ...##
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Eigenvalues of Cayley graphs
[article]

2019
*
arXiv
*
pre-print

A family of Cayley graphs on

arXiv:1809.09829v2
fatcat:57hq3pcaurhybgfuzzh6qi74nu
*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. ...##
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Invariant semidefinite programs
[article]

2010
*
arXiv
*
pre-print

In the last years many results in the area of semidefinite programming were obtained

arXiv:1007.2905v2
fatcat:nhbqxf5jqjgyrjfr3qnk4t4rie
*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. ...##
###
Invariant Semidefinite Programs
[chapter]

2011
*
International Series in Operations Research and Management Science
*

This chapter provides the reader with the necessary background

doi:10.1007/978-1-4614-0769-0_9
fatcat:vnun3jk5f5hwnjabutyxmtsesu
*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). ...##
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Index

1999
*
Linear Algebra and its Applications
*

Sanchez-Giralda, Feedback

doi:10.1016/s0024-3795(99)00243-8
fatcat:xz5sbc3lzngydobg3ln6xp2cum
*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. ...
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