MacWilliams Identities for Linear Codes over Finite Frobenius Rings

2001 ◽  
pp. 276-292 ◽  
Author(s):  
Thomas Honold ◽  
Ivan Landjev
Keyword(s):  
Linear Codes ◽  
Frobenius Rings ◽  
Macwilliams Identities ◽  
Finite Frobenius Rings

scholarly journals FINITE QUASI-FROBENIUS MODULES AND LINEAR CODES

2004 ◽  
Vol 03 (03) ◽  
pp. 247-272 ◽  
Author(s):  
MARCUS GREFERATH ◽  
ALEXANDR NECHAEV ◽  
ROBERT WISBAUER
Keyword(s):  
Finite Fields ◽  
Linear Codes ◽  
General Setting ◽  
Commutative Rings ◽  
Finite Ring ◽  
Weight Enumerators ◽  
Frobenius Rings ◽  
Finite Frobenius Rings

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

scholarly journals New bounds for codes over finite Frobenius rings

2010 ◽  
Vol 57 (2) ◽  
pp. 169-179 ◽  
Author(s):  
Eimear Byrne ◽  
Marcus Greferath ◽  
Axel Kohnert ◽  
Vitaly Skachek
Keyword(s):  
Frobenius Rings ◽  
Finite Frobenius Rings

THE EQUIVALENT IDENTITIES OF THE MACWILLIAMS IDENTITIES FOR LINEAR CODES

2015 ◽  
Vol 76 (1) ◽  
Author(s):  
Bao Xiaomin
Keyword(s):  
Linear Codes ◽  
Macwilliams Identity ◽  
Macwilliams Identities

We use derivatives to prove the equivalences between MacWilliams identity and its four equivalent forms, and present new interpretations for the four equivalent forms.

ON PLOTKIN-OPTIMAL CODES OVER FINITE FROBENIUS RINGS

2006 ◽  
Vol 05 (06) ◽  
pp. 799-815 ◽  
Author(s):  
MARCUS GREFERATH ◽  
GARY McGUIRE ◽  
MICHAEL E. O'SULLIVAN
Keyword(s):  
Hadamard Matrices ◽  
Interesting Property ◽  
Optimal Codes ◽  
Frobenius Ring ◽  
Frobenius Rings ◽  
Plotkin Bound ◽  
Homogeneous Weight ◽  
Finite Frobenius Rings ◽  
Finite Frobenius Ring ◽  
The Relationship

We study the Plotkin bound for codes over a finite Frobenius ring R equipped with the homogeneous weight. We show that for codes meeting the Plotkin bound, the distribution on R induced by projection onto a coordinate has an interesting property. We present several constructions of codes meeting the Plotkin bound and of Plotkin-optimal codes. We also investigate the relationship between Butson–Hadamard matrices and codes over R meeting the Plotkin bound.

scholarly journals Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉

2018 ◽  
Vol 16 (1) ◽  
pp. 490-497
Author(s):  
Xiying Zheng ◽  
Bo Kong
Keyword(s):  
Linear Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Macwilliams Identities

AbstractIn this paper, we study linear codes over ring Rk = 𝔽pm[u1, u2,⋯,uk]/〈$\begin{array}{} u^{2}_{i} \end{array} $ = ui, uiuj = ujui〉 where k ≥ 1 and 1 ≤ i, j ≤ k. We define a Gray map from $\begin{array}{} R_{k}^n\,\,\text{to}\,\,{\mathbb F}_{p^m}^{2^kn} \end{array} $ and give the generator polynomials of constacyclic codes over Rk. We also study the MacWilliams identities of linear codes over Rk.

scholarly journals On the MacWilliams identities for linear codes

1988 ◽  
Vol 107 ◽  
pp. 181-189 ◽  
Author(s):  
Richard A. Brualdi ◽  
Vera S. Pless ◽  
Janet S. Beissinger
Keyword(s):  
Linear Codes ◽  
Macwilliams Identities
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