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Code equivalence characterizes finite Frobenius rings
2007
Proceedings of the American Mathematical Society
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In this paper we show that finite rings for which the code equivalence theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. ...
The author thanks several anonymous referees for their helpful suggestions. The author also thanks E. S. Moore for unwavering support. ...
Linear codes over a finite Frobenius ring also have the extension property [16] . ...
doi:10.1090/s0002-9939-07-09164-2
fatcat:dl4qbw44a5c3pbvplzimcxdwoa
The Extension Theorem with Respect to Symmetrized Weight Compositions
[chapter]
2015
Coding Theory and Applications
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We will say that an alphabet A satisfies the extension property with respect to a weight w if every linear isomorphism between two linear codes in A n that preserves w extends to a monomial transformation ...
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. ...
The Extension Theorem for symmetrized weight compositions was proved for linear codes over finite fields in [5] , over finite Frobenius rings in [15] , and over Frobenius bimodule alphabets in [19] ...
doi:10.1007/978-3-319-17296-5_18
dblp:conf/icmcta/ElGaremMW14
fatcat:dwmsz3wht5f7veb2uk3hwx72qm
On Counting Subring-Subcodes of Free Linear Codes Over Finite Principal Ideal Rings
[article]
2016
arXiv
pre-print
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Let R be a finite principal ideal ring and S the Galois extension of R of degree m. ...
For k and k_0, positive integers we determine the number of free S-linear codes B of length l with the property k = rank_S(B) and k_0 = rank_R (B∩ R^l). ...
Counting codes over finite PIRs In this section, we extend our results to linear codes over finite principal ideal rings (PIRs). ...
arXiv:1612.02213v1
fatcat:3hmj4l2bsfhfdcc5ltkvhg3xsa
MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings
[article]
2013
arXiv
pre-print
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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. ...
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 ...
the Extension Theorem is true for linear codes over finite Frobenius rings when using the homogeneous weight. ...
arXiv:1309.3292v1
fatcat:owdhd3k6rbdyhctbmvqjv2ryfa
Galois Correspondence on Linear Codes over Finite Chain Rings
[article]
2016
arXiv
pre-print
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Given S|R a finite Galois extension of finite chain rings and B an S-linear code we define two Galois operators, the closure operator and the interior operator. ...
As applications some improvements of upper and lower bounds for the rank of the restriction and trace code are given and some applications to S-linear cyclic codes are shown. ...
Finally in Section 4 we describe linear cyclic codes over finite chain rings as the restriction of a linear cyclic code over a Galois extension of a finite chain ring. ...
arXiv:1602.01242v2
fatcat:qpw3im4hufeyphpinvnfdxhz4m
FOUNDATIONS OF LINEAR CODES DEFINED OVER FINITE MODULES: THE EXTENSION THEOREM AND THE MACWILLIAMS IDENTITIES
2009
Codes Over Rings
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This paper discusses the foundations of the theory of linear codes defined over finite modules. Two topics are examined in depth: the extension theorem and the MacWilliams identities. ...
Both of these topics were studied originally by MacWilliams in the context of linear codes defined over finite fields. WSPC -Proceedings Trim Size: 9in x 6in wood-ankara-9x6-rev-051909 6 2.3. ...
I thank the referee for a number of valuable suggestions for improving this paper and Ryan Schwiebert for pointing out an error in an earlier form of Lemma 5.2. ...
doi:10.1142/9789812837691_0004
fatcat:lc2qhrx7yra5halm7h4kaf6voq
Finite-Ring Combinatorics and MacWilliams' Equivalence Theorem
2000
Journal of combinatorial theory. Series A
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Constantinescu's concept of homogeneous weights on arbitrary finite rings and prove MacWilliams' equivalence theorem to hold with respect to these weights for all finite Frobenius rings. ...
MacWilliams proved that Hamming isometries between linear codes extend to monomial transformations. This theorem has recently been generalized by J. ...
Wood [15 17] , the former proving an extension theorem for homogeneous weight functions over integer residue rings, the latter developing extension results for the up to now most general class of rings ...
doi:10.1006/jcta.1999.3033
fatcat:h6hbusw4bbhydjdrjzaym5mwja
On the Lattice of Cyclic Linear Codes Over Finite Chain Rings
[article]
2017
arXiv
pre-print
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Let R be a commutative finite chain ring of invariants (q,s). ...
The lattice (Cy(R,ℓ), +, ∩) of cyclic R-linear codes of length ℓ, is investigated. A lower bound on the Hamming distance of cyclic R-linear codes of length ℓ, is established. ...
LINEAR CODES OVER FINITE CHAIN RINGS For this section, R is a finite chain ring of invariants (q, s ), and θ is a generator of maximal ideal J(R). ...
arXiv:1701.08740v1
fatcat:j3hxq2nh7jgy3oib4q2fgx35im
MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings
2014
Journal of combinatorial theory. Series A
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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 . ...
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 ...
the Extension Theorem is true for linear codes over finite Frobenius rings when using the homogeneous weight. ...
doi:10.1016/j.jcta.2014.03.005
fatcat:jjxgvpmomnejhniulj43cafcqa
On the Equivalence of Codes over Finite Rings
2004
Applicable Algebra in Engineering, Communication and Computing
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It is known that if a finite ring R is Frobenius then equivalences of linear codes over R are always monomial transformations. ...
Namely, it is shown that for every finite ring R which is a direct sum of local and homogeneous semilocal subrings, if every Hamming-weight preserving R-linear transformation of a code C 1 onto a code ...
The authors wish to thank the referees for a very careful reading of the original version of this manuscript and their valuable suggestions to improve it. ...
doi:10.1007/s00200-004-0149-5
fatcat:wvjordxcczgonjbeoilyq2yljm
Characteristics of invariant weights related to code equivalence over rings
2012
Designs, Codes and Cryptography
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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. ...
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. ...
A character theoretic proof of this Extension Theorem in [16] led to a generalisation of this theorem for codes over finite Frobenius rings in [17] . ...
doi:10.1007/s10623-012-9671-9
fatcat:ld6daw4jhncpzhuy46bgamte2a
Characteristics of Invariant Weights Related to Code Equivalence over Rings
[article]
2011
arXiv
pre-print
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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. ...
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. ...
A character theoretic proof of this Extension Theorem in [16] led to a generalisation of this theorem for codes over finite Frobenius rings in [17] . ...
arXiv:1110.1538v1
fatcat:iweebpwj7rf6vnw7shirpiski4
Theory and applications of linearized multivariate skew polynomials
[article]
2021
arXiv
pre-print
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2020 pre-print arXiv:2001.01273v2 fatcat:5skkj7pqknbbpdvxa4kbyxy47eOther Versions
Such polynomials are right linear over the corresponding centralizer and generalize linearized polynomial rings over finite fields, group rings or differential polynomial rings. ...
P-Galois extensions of division rings are then introduced, which generalize classical (finite) Galois extensions. ...
Recently, a linearized notion of Reed-Muller codes over finite extensions of fields was introduced in [4] . ...
arXiv:2001.01273v3
fatcat:5fhoxnhyh5a5pdb3nsvyyx7rsu
Relative one-weight linear codes
2012
Designs, Codes and Cryptography
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Relative one-weight linear codes were introduced by Liu and Chen over finite fields. ...
These codes can be defined just as simply for egalitarian and homogeneous weights over Frobenius bimodule alphabets. ...
for this paper was conducted. ...
doi:10.1007/s10623-012-9769-0
fatcat:oaelohfzfzhxpiuguh3rvonnhu
On the Hamming distance of linear codes over a finite chain ring
2000
IEEE Transactions on Information Theory
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a Gray map of linear codes over Z 4 ; see [6] [17] , for Galois rings in [7] and more generally for any finite chain ring in [12] . ...
Most of the well-known algebraic decoding algorithms for linear codes over finite fields use Hamming distance. Some of these algorithms can be generalised to linear codes over finite rings. ...
doi:10.1109/18.841186
fatcat:k46kgpop6nbdnf5t5jusvwg5hm
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