Abstract
Consider the set of all error-correcting block codes over a fixed alphabet with q letters. It determines a recursively enumerable set of points in the unit square with coordinates (R,δ):= (relative transmission rate, relative minimal distance). Limit points of this set form a closed subset, defined by R ≤ α q (δ), where α q (δ) is a continuous decreasing function called asymptotic bound. Its existence was proved by the author in 1981, but all attempts to find an explicit formula for it so far failed.
In this note I consider the question whether this function is computable in the sense of constructive mathematics, and discuss some arguments suggesting that the answer might be negative.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Barg, A., Forney, G.D.: Random codes: minimum distances and error exponents. IEEE Transactions on Information Theory 48(9), 2568–2573 (2002)
Brattka, V.: Plottable real functions and the computable graph theorem. SIAM J. Comput. 38(1), 303–328 (2008)
Brattka, V., Miller, J.S., Nies, A.: Randomness and differentiability. arXiv:1104.4456
Brattka, V., Preser, G.: Computability on subsets of metric spaces. Theoretical Computer Science 305, 43–76 (2003)
Brattka, V., Weihrauch, K.: Computability on subsets of Euclidean space I: closed and compact subsets. Theoretical Computer Science 219, 65–93 (1999)
Braverman, M., Cook, S.: Computing over the reals: foundations for scientific computing. Notices AMS 53(3), 318–329 (2006)
Braverman, M., Yampolsky, M.: Computability of Julia sets. Moscow Math. Journ. 8(2), 185–231 (2008)
Calude, C.S., Hertling, P., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin Ω numbers. Theor. Comp. Sci. 255, 125–149 (2001)
Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles., I–III. C. R. Ac. Sci. Paris 240, 2478–2480, 241, 13–14, 151–153 (1955)
Manin, Y.I.: What is the maximum number of points on a curve over \(\textbf{F}_2\)? J. Fac. Sci. Tokyo, IA 28, 715–720 (1981)
Manin, Y.I.: Renormalization and computation I: motivation and background. Preprint math. QA/0904.4921
Manin, Y.I.: Renormalization and Computation II: Time Cut-off and the Halting Problem. Preprint math. QA/0908.3430
Manin, Y.I., Marcolli, M.: Error-correcting codes and phase transitions. arXiv:0910.5135
Manin, Y.I., Vladut, S.G.: Linear codes and modular curves. J. Soviet Math. 30, 2611–2643 (1985)
Tsfasman, M.A., Vladut, S.G.: Algebraic-geometric Codes. Kluwer (1991)
Vladut, S.G., Nogin, D.Y., Tsfasman, M.A.: Algebraic Geometric Codes: Basic Notions. Mathematical Surveys and Monographs, vol. 139. American Mathematical Society, Providence (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Manin, Y.I. (2012). A Computability Challenge: Asymptotic Bounds for Error-Correcting Codes. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-27654-5_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27653-8
Online ISBN: 978-3-642-27654-5
eBook Packages: Computer ScienceComputer Science (R0)