Abstract
The function s(u) arising in the study of long primitive BCH codes over GF(q) is reviewed. The set of points 0 < u ≤ 1 such that q ku has a modulo 1 representation in the interval [a, 1] for every integer k ≥ 0 is shown to have Hausdorff dimension s(a) for every 0 ≤ a ≤ 1. Berlekamp's conjecture on the dimension of a set of points at which s fails to be differentiable is also proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berlekamp, E.R. 1984. Algebraic Coding Theory (revised 1984 edition). Laguna Hills, CA: Aegean Park.
Berlekamp, E.R. 1972. Long primitive binary BCH codes have distance d ~ 2n In R -1/log n... IEEE Trans. Inform. Theory, vol. IT-18:415–426.
Kolmogorov, A.N., and Fomin, S.V. 1975. Introductory Real Analysis. (R.A. Silverman, Ed.). New York: Dover.
Falconer, K.J. 1986. The Geometry of Fractal Sets. Cambridge: Cambridge University.
Author information
Authors and Affiliations
Additional information
Communicated by S. Vanstone
Rights and permissions
About this article
Cite this article
Peterson, D.J. Fractal properties of the singular function s(u) . Des Codes Crypt 1, 133–139 (1991). https://doi.org/10.1007/BF00157617
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00157617