Abstract
We study weight distributions of shifts of codes from a well-known family: the 3-error-correcting binary nonlinear Goethals-like codes of length n = 2m, where m ≥ 6 is even. These codes have covering radius ρ = 6. We know the weight distribution of any shift of weight i = 1, 2, 3, 5, or 6. For a shift of weight 4, the weight distribution is uniquely defined by the number of leaders in this shift, i.e., the number of vectors of weight 4. We also consider the weight distribution of shifts of codes with minimum distance 7 obtained by deleting any one position of a Goethals-like code of length n.
Similar content being viewed by others
REFERENCES
Helleseth, T. and Zinoviev, V.A., On Coset Weight Distributions of the Z4-Linear Goethals Codes, IEEE Trans. Inf. Theory, 2001, vol. 49, no. 5, pp. 1758–1772.
Helleseth, T. and Zinoviev, V., Codes with the Same Shift Weight Distributions as the Z4-Linear Goethals Codes, IEEE Trans. Inf. Theory, 2001, vol. 49, no. 4, pp. 1589–1596.
Goethals, J.-M., Two Dual Families of Nonlinear Binary Codes, Electron. Lett., 1974, vol. 10, no. 23, pp. 471–472.
Goethals, J.-M., Nonlinear Codes Defined by Quadratic Forms over GF(2), Inf. Control, 1976, vol. 31, no. 1, pp. 43–74.
Charpin, P. and Zinoviev, V.A., On Coset Weight Distributions of the 3-Error-Correcting BCH Codes, SIAM J. Discrete Math., 1997, vol. 10, no. 1, pp. 128–145.
Delsarte P., Four Fundamental Parameters of a Code and Their Combinatorial Signi_cance, Inf. Control, 1973, vol. 23, no. 5, pp. 407–438.
Bassalygo, L.A., Zaitsev, G.V., and Zinoviev, V.A., Uniformly Packed Codes, Probl. Peredachi Inf., 1974, vol. 10, no. 1, pp. 9–14[Probl. Inf. Trans. (Engl. Transl.), 1974, vol. 10, no. 1, pp. 6–9].
Goethals, J.-M. and van Tilborg, H.C.A., Uniformly Packed Codes, Philips Res. Rep., 1975, vol. 30, pp. 9–36.
Semakov, N.V., Zinoviev, V.A., and Zaitsev, G.V., Uniformly Packed Codes, Probl. Peredachi Inf., 1971, vol. 7, no. 1, pp. 38–50[Probl. Inf. Trans. (Engl. Transl.), 1971, vol. 7, no. 1, pp. 30–39].
van Tilborg, H.C.A., Uniformly Packed Codes, PhD Thesis, Tech. Univ. Eindhoven, 1976.
Bassalygo, L.A. and Zinoviev, V.A., Remark on Uniformly Packed Codes, Probl. Peredachi Inf., 1977, vol. 13, no. 3, pp. 22–25[Probl. Inf. Trans. (Engl. Transl.), 1977, vol. 13, no. 3, pp. 178–180].
Brualdi, R.A. and Pless, V.S., Orphans of the First Order Reed–Muller Codes, IEEE Trans. Inf. Theory, 1990, vol. 36, no. 2, pp. 399–401.
Helleseth, T. and Kumar, P.V., The Algebraic Decoding of the ℤ4-Linear Goethals Code, IEEE Trans. Inf. Theory, 1995, vol. 41, no. 6, pp. 2040–2048.
Sloane, N.J.A. and Dick, R.J., On the Enumeration of Shifts of First Order Reed–Muller Codes, Proc. IEEE Int. Conf. on Communications, Montreal, 1971, New York: IEEE Press, 1971, vol. 7, pp. 36-2 to 36-6.
MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz', 1979.
Helleseth, T., Kumar, P.V., and Shanbhag, A., Codes with the Same Weight Distribution as the Goethals Codes and Delsarte–Goethals Codes, Designs Codes Crypt., 1996, vol. 9, no. 2, pp. 257–266.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zinoviev, V.A., Helleseth, T. On Weight Distributions of Shifts of Goethals-like Codes. Problems of Information Transmission 40, 118–134 (2004). https://doi.org/10.1023/B:PRIT.0000043926.60991.7e
Issue Date:
DOI: https://doi.org/10.1023/B:PRIT.0000043926.60991.7e