{"title":"Pontrjagin Duality and Codes over Finite Commutative Rings","authors":"Khalid Abdelmoumen, Mustapha Najmeddine, Hussain Ben-Azza","volume":56,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1181,"pagesEnd":1186,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/4423","abstract":"We present linear codes over finite commutative rings\r\nwhich are not necessarily Frobenius. We treat the notion of syndrome\r\ndecoding by using Pontrjagin duality. We also give a version of Delsarte-s\r\ntheorem over rings relating trace codes and subring subcodes.","references":"[1] J. Wood J. Duality for modules over finite rings and applications to coding\r\ntheory. Amer. J. Math. 121, pp. 555-575 (1999).\r\n[2] J. Wood J. Foundations of Linear Codes defined over Finite Modules\r\n: The extension Theorem and the MacWilliams Identities. In - Codes\r\nover Rings, Proceedings of the CIMPA Summer School, Ankara, Turkey,\r\n18-29 August 2008, Patrick Sol, editor-, Series on Coding Theory and\r\nCrytology, Vol. 6, World Scientific, Singapore, 2009, pp. 124-190.\r\n[3] C. W. Curtis and I. Reiner. Representation Theory of Finite Groups and\r\nAssociative Algebras. Interscience Publishers, 1962.\r\n[4] H. Stichtenoth. Algebraic Function Fields and Codes. Springer, 1993.\r\n[5] S. T. Dougherty and H. Liu, Independence of vectors in codes over rings,\r\nDesigns, Codes and Cryptography, Volume 51, Number 1, 55-68, 2009.\r\n[6] M. F. Atiyah and I. G. Macdonald. Introduction to commutative Algebra.\r\nAddison-Wesley, 1969.\r\n[7] W. Rudin. Fourier Analysis on Groups, Wiley-Interscience, 1990.\r\n[8] M. Giorgetti and A. Previtali. Galois invariance, traces codes and subfield\r\nsubcodes. Finite Fields and Their Applications 16(2): 96-99 (2010).\r\n[9] B. A. McDonald. Finite Rings with Identity. Marcel Dekker, 1974.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 56, 2011"}