Pontrjagin Duality and Codes over Finite Commutative Rings
We present linear codes over finite commutative rings
which are not necessarily Frobenius. We treat the notion of syndrome
decoding by using Pontrjagin duality. We also give a version of Delsarte-s
theorem over rings relating trace codes and subring subcodes.
[1] J. Wood J. Duality for modules over finite rings and applications to coding
theory. Amer. J. Math. 121, pp. 555-575 (1999).
[2] J. Wood J. Foundations of Linear Codes defined over Finite Modules
: The extension Theorem and the MacWilliams Identities. In - Codes
over Rings, Proceedings of the CIMPA Summer School, Ankara, Turkey,
18-29 August 2008, Patrick Sol, editor-, Series on Coding Theory and
Crytology, Vol. 6, World Scientific, Singapore, 2009, pp. 124-190.
[3] C. W. Curtis and I. Reiner. Representation Theory of Finite Groups and
Associative Algebras. Interscience Publishers, 1962.
[4] H. Stichtenoth. Algebraic Function Fields and Codes. Springer, 1993.
[5] S. T. Dougherty and H. Liu, Independence of vectors in codes over rings,
Designs, Codes and Cryptography, Volume 51, Number 1, 55-68, 2009.
[6] M. F. Atiyah and I. G. Macdonald. Introduction to commutative Algebra.
Addison-Wesley, 1969.
[7] W. Rudin. Fourier Analysis on Groups, Wiley-Interscience, 1990.
[8] M. Giorgetti and A. Previtali. Galois invariance, traces codes and subfield
subcodes. Finite Fields and Their Applications 16(2): 96-99 (2010).
[9] B. A. McDonald. Finite Rings with Identity. Marcel Dekker, 1974.
[1] J. Wood J. Duality for modules over finite rings and applications to coding
theory. Amer. J. Math. 121, pp. 555-575 (1999).
[2] J. Wood J. Foundations of Linear Codes defined over Finite Modules
: The extension Theorem and the MacWilliams Identities. In - Codes
over Rings, Proceedings of the CIMPA Summer School, Ankara, Turkey,
18-29 August 2008, Patrick Sol, editor-, Series on Coding Theory and
Crytology, Vol. 6, World Scientific, Singapore, 2009, pp. 124-190.
[3] C. W. Curtis and I. Reiner. Representation Theory of Finite Groups and
Associative Algebras. Interscience Publishers, 1962.
[4] H. Stichtenoth. Algebraic Function Fields and Codes. Springer, 1993.
[5] S. T. Dougherty and H. Liu, Independence of vectors in codes over rings,
Designs, Codes and Cryptography, Volume 51, Number 1, 55-68, 2009.
[6] M. F. Atiyah and I. G. Macdonald. Introduction to commutative Algebra.
Addison-Wesley, 1969.
[7] W. Rudin. Fourier Analysis on Groups, Wiley-Interscience, 1990.
[8] M. Giorgetti and A. Previtali. Galois invariance, traces codes and subfield
subcodes. Finite Fields and Their Applications 16(2): 96-99 (2010).
[9] B. A. McDonald. Finite Rings with Identity. Marcel Dekker, 1974.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53533", author = "Khalid Abdelmoumen and Mustapha Najmeddine and Hussain Ben-Azza", title = "Pontrjagin Duality and Codes over Finite Commutative Rings", abstract = "We present linear codes over finite commutative rings
which are not necessarily Frobenius. We treat the notion of syndrome
decoding by using Pontrjagin duality. We also give a version of Delsarte-s
theorem over rings relating trace codes and subring subcodes.", keywords = "Codes, Finite Rings, Pontrjagin Duality, Trace Codes.", volume = "5", number = "8", pages = "1219-5", }