Abstract
We examine the well-known problem of determining the capacity of multi-dimensional run-length-limited constrained systems. By recasting the problem, which is essentially a combinatorial counting problem, into a probabilistic setting, we are able to derive new lower and upper bounds on the capacity of (0,k)-RLL systems. These bounds are better than all previously-known bounds for k ≥ 2, and are even tight asymptotically. Thus, we settle the open question: what is the rate at which the capacity of (0,k)-RLL systems converges to 1 as k → ∞? While doing so, we also provide the first ever non-trivial upper bound on the capacity of general (d,k)-RLL systems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Calkin, N., Wilf, H.: The number of independent sets in the grid graph. SIAM J. Discrete Math. 11, 54–60 (1998)
Halevy, S., Chen, J., Roth, R.M., Siegel, P.H., Wolf, J.K.: Improved bit-stuffing bounds on two-dimensional constraints. IEEE Trans. on Inform. Theory 50(5), 824–838 (2004)
Ito, H., Kato, A., Nagy, Z., Zeger, K.: Zero capacity region of multidimensional run length constraints. Elec. J. of Comb. 6 (1999)
Janson, S.: Poisson approximation for large deviations. Random Structures and Algorithms 1, 221–230 (1990)
Kato, A., Zeger, K.: On the capacity of two-dimensional run-length constrained channels. IEEE Trans. on Inform. Theory 45, 1527–1540 (1999)
Kleitman, D.J.: Families of non-disjoint subsets. J. Combin. Theory 1, 153–155 (1966)
Marcus, B.H., Siegel, P.H., Wolf, J.K.: Finite-state modulation codes for data storage. IEEE J. Select. Areas Commun. 10, 5–37 (1992)
Marcus, B.H., Roth, R.M., Siegel, P.H.: Constrained systems and coding for recording channels. In: Pless, V.S., Huffman, W.C. (eds.), Elsevier, Amsterdam (1998)
Nagy, Z., Zeger, K.: Capacity bounds for the three-dimensional (0,1) run length limited channel. IEEE Trans. on Inform. Theory 46(3), 1030–1033 (2000)
Shannon, C.E.: A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948)
Shannon, C.E.: A mathematical theory of communication. Bell System Technical Journal 27, 623–656 (1948)
Talyansky, R.: Coding for two-dimensional constraints. M.Sc. thesis, Computer Science Dep., Technion – Israel Institute of Technology, Haifa, Israel (1997) (in Hebrew)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schwartz, M., Vardy, A. (2006). New Bounds on the Capacity of Multi-dimensional RLL-Constrained Systems. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_22
Download citation
DOI: https://doi.org/10.1007/11617983_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31423-3
Online ISBN: 978-3-540-31424-0
eBook Packages: Computer ScienceComputer Science (R0)