Abstract
Cyclic codes with symbols from a residue class integer ringZ m are characterized in terms of the discrete Fourier transform (DFT) of codewords defined over an appropriate extension ring ofZ m . It is shown that a cyclic code of length n overZ m ,n relatively prime tom, consists ofn-tuples overZ m having a specified set of DFT coefficients from the elements of an ideal of a subring of the extension ring. Whenm is equal to a product of distinct primes every cyclic code overZ m has an idempotent generator and it is shown that the idempotent generators can be easily identified in the transform domain. The dual code pairs overZ m are characterized in the transform domain for cyclic codes. Necessary and sufficient conditions for the existence of self-dual codes overZ m are obtained and nonexistence of self-dual codes for certain values ofm is proved.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Berlekamp, E. R.: Algebraic Coding Theory. New York: McGraw Hill 1968
Blahut, R. E.: Theory and Practice of Error Control Codes. California: Addision-Wesley 1983
Blake, I. F.: Codes over certain rings. Inform. Control20, 296–404 (1972)
Blake, I. F.: Codes over integer residue rings. Inform. Control29, 295–300 (1975)
Britten, J. D., Lemire, E. W.: A structure theorem for rings supporting a discrete Fourier transform. SIAM J. Appl. Math.41, 222–226 (1981)
Chiang, J., Wolf, J. K.: On channels and codes for the Lee metric. Inform. Control19, 159–173 (1971)
Delsarte, P.: Bounds for unrestricted codes by linear programming. Philips Research Dev. J.27, 272–289 (1972)
Dubios, E., Venetsanopoulos, A. N.: The discrete Fourier transform over finite rings with application to fast convolution. IEEE Trans. Computers,C-27, 586–593 (1978)
Madhusudhana, H. S.: On Abelian codes which are closed under cyclic shifts, M. Tech Thesis, Indian Institute of Tech. Kanpur (India), 1987
Martens, J. B., Vanwormhoudt, M. C.: Convolution using a conjugate symmetry property for number theoretic transforms over rings of regular integers. IEEE Trans. ASSP, ASSP-31, 1121–1124 (1983)
Massey, J. L., Mittelholzer, T. M.: Convolutional codes over rings. Proceedings of the Fourth Joint Swedish-USSR Int. Workshop in Information Theory,27, Gotland, Sweden, 1989
McDonald, B. R.: Finite Rings with identity, New York: Marcel-Decker 1974
Murakami, H., Reed, I. S., Welch, L. R.: A tranform decider for Reed-Solomon codes in multiple user communication systems. IEEE Trans. Inform. TheoryIT-23, 1745–1753 (1977)
Nemirovskiy, E. E.: Codes on residue class rings with multi-freequency phase telegraphy. Radiotechnika i electronika9, 1745–1753 (1984)
Prithi Shankar: On BCH codes over arbitrary integer rings. IEEE Trans. Inform. Theory,IT-25, 480–483 (1979)
Reeds, J. A., Sloane, N. J. A.: Shift Register Synthesis (modulom). SIAM J. Computing, 505–513 (1985)
Spiegel, E.: Codes overZ m . Inform. Control35, 48–52 (1977)
Spiegel, E.: Codes overZ m -Revisited. Inform. Control37, 100–104 (1978)
Sundar Rajan, B., Siddiqi, M. U.: Transform decoding of BCH codes overZ m (To appear in Internat. J. Electronics)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rajan, B.S., Siddiqi, M.U. Transform domain characterization of cyclic codes overZ m . AAECC 5, 261–275 (1994). https://doi.org/10.1007/BF01225641
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01225641