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Code equivalence characterizes finite Frobenius rings

2007
*
Proceedings of the American Mathematical Society
*

In this paper we show that

doi:10.1090/s0002-9939-07-09164-2
fatcat:dl4qbw44a5c3pbvplzimcxdwoa
*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] . ...##
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The Extension Theorem with Respect to Symmetrized Weight Compositions
[chapter]

2015
*
Coding Theory and Applications
*

We will say that an alphabet A satisfies the

doi:10.1007/978-3-319-17296-5_18
dblp:conf/icmcta/ElGaremMW14
fatcat:dwmsz3wht5f7veb2uk3hwx72qm
*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] ...##
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On Counting Subring-Subcodes of Free Linear Codes Over Finite Principal Ideal Rings
[article]

2016
*
arXiv
*
pre-print

Let R be a

arXiv:1612.02213v1
fatcat:3hmj4l2bsfhfdcc5ltkvhg3xsa
*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). ...##
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MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings
[article]

2013
*
arXiv
*
pre-print

A

arXiv:1309.3292v1
fatcat:owdhd3k6rbdyhctbmvqjv2ryfa
*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. ...##
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Galois Correspondence on Linear Codes over Finite Chain Rings
[article]

2016
*
arXiv
*
pre-print

Given S|R a

arXiv:1602.01242v2
fatcat:qpw3im4hufeyphpinvnfdxhz4m
*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*. ...##
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FOUNDATIONS OF LINEAR CODES DEFINED OVER FINITE MODULES: THE EXTENSION THEOREM AND THE MACWILLIAMS IDENTITIES

2009
*
Codes Over Rings
*

This paper discusses the foundations of the theory of

doi:10.1142/9789812837691_0004
fatcat:lc2qhrx7yra5halm7h4kaf6voq
*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. ...##
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Finite-Ring Combinatorics and MacWilliams' Equivalence Theorem

2000
*
Journal of combinatorial theory. Series A
*

Constantinescu's concept of homogeneous weights on arbitrary

doi:10.1006/jcta.1999.3033
fatcat:h6hbusw4bbhydjdrjzaym5mwja
*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*...##
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On the Lattice of Cyclic Linear Codes Over Finite Chain Rings
[article]

2017
*
arXiv
*
pre-print

Let R be a commutative

arXiv:1701.08740v1
fatcat:j3hxq2nh7jgy3oib4q2fgx35im
*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). ...##
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MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

2014
*
Journal of combinatorial theory. Series A
*

A

doi:10.1016/j.jcta.2014.03.005
fatcat:jjxgvpmomnejhniulj43cafcqa
*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. ...##
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On the Equivalence of Codes over Finite Rings

2004
*
Applicable Algebra in Engineering, Communication and Computing
*

It is known that if a

doi:10.1007/s00200-004-0149-5
fatcat:wvjordxcczgonjbeoilyq2yljm
*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. ...##
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Characteristics of invariant weights related to code equivalence over rings

2012
*
Designs, Codes and Cryptography
*

This

doi:10.1007/s10623-012-9671-9
fatcat:ld6daw4jhncpzhuy46bgamte2a
*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] . ...##
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Characteristics of Invariant Weights Related to Code Equivalence over Rings
[article]

2011
*
arXiv
*
pre-print

This

arXiv:1110.1538v1
fatcat:iweebpwj7rf6vnw7shirpiski4
*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] . ...##
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Theory and applications of linearized multivariate skew polynomials
[article]

2021
*
arXiv
*
pre-print

Such polynomials are right

arXiv:2001.01273v3
fatcat:5fhoxnhyh5a5pdb3nsvyyx7rsu
*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] . ...##
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Relative one-weight linear codes

2012
*
Designs, Codes and Cryptography
*

Relative one-weight

doi:10.1007/s10623-012-9769-0
fatcat:oaelohfzfzhxpiuguh3rvonnhu
*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. ...##
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On the Hamming distance of linear codes over a finite chain ring

2000
*
IEEE Transactions on Information Theory
*

a Gray map of

doi:10.1109/18.841186
fatcat:k46kgpop6nbdnf5t5jusvwg5hm
*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*. ...
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