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Code equivalence characterizes finite Frobenius rings

Jay A. Wood
2007 Proceedings of the American Mathematical Society  
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]

Noha ElGarem, Nefertiti Megahed, Jay A. Wood
2015 Coding Theory and Applications  
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]

Ramakrishna Bandi and Alexandre Fotue Tabue and Edgar Martínez-Moro
2016 arXiv   pre-print
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]

Marcus Greferath, Thomas Honold, Cathy Mc Fadden, Jay A. Wood, and Jens Zumbrägel
2013 arXiv   pre-print
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]

A. Fotue Tabue and E. Martínez-Moro and C. Mouaha
2016 arXiv   pre-print
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

Jay A. Wood
2009 Codes Over Rings  
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

M. Greferath, S.E. Schmidt
2000 Journal of combinatorial theory. Series A  
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]

Alexandre Fotue Tabue, Christophe Mouaha
2017 arXiv   pre-print
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

Marcus Greferath, Thomas Honold, Cathy Mc Fadden, Jay A. Wood, Jens Zumbrägel
2014 Journal of combinatorial theory. Series A  
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

Hai Quang Dinh, Sergio R. L�pez-Permouth
2004 Applicable Algebra in Engineering, Communication and Computing  
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

Marcus Greferath, Cathy Mc Fadden, Jens Zumbrägel
2012 Designs, Codes and Cryptography  
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]

Marcus Greferath, Cathy Mc Fadden, Jens Zumbrägel
2011 arXiv   pre-print
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]

Umberto Martínez-Peñas
2021 arXiv   pre-print
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

Jay A. Wood
2012 Designs, Codes and Cryptography  
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

G.H. Norton, A. Salagean
2000 IEEE Transactions on Information Theory  
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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