TY - JOUR

T1 - Constrained codes as networks of relations

AU - Schwartz, Mosche

AU - Bruck, Jehoshua

N1 - Funding Information:
Manuscript received August 5, 2007; revised January 6, 2008. This work was supported in part by the Caltech Lee Center for Advanced Networking. M. Schwartz was with the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, USA. He is now with the Department of Electrical and Computer Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel (e-mail: [email protected]). J. Bruck is with the Department of Electrical Engineering, California Institute ofTechnology,Pasadena,CA91125USA(e-mail:[email protected]). Communicated by T. J. Richardson, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2008.920245

PY - 2008/5/1

Y1 - 2008/5/1

N2 - We address the well-known problem of determining the capacity of constrained coding systems. While the one-dimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a two-dimensional constrained coding system is still an elusive research challenge. The only known exception in the two-dimensional case is an exact (however, not rigorous) solution to the (1,∞) -run-length limited (RLL) system on the hexagonal lattice. Furthermore, only exponential-time algorithms are known for the related problem of counting the exact number of constrained two-dimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a two-dimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graph-theoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique, we derive a closed-form solution to the capacity related to the Path-Cover constraint in a two-dimensional triangular array (the resulting calculated capacity is 0.72399217...). Path-Cover is a generalization of the well known one-dimensional (0.1) b-RLL constraint for which the capacity is known to be 0.69424....

AB - We address the well-known problem of determining the capacity of constrained coding systems. While the one-dimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a two-dimensional constrained coding system is still an elusive research challenge. The only known exception in the two-dimensional case is an exact (however, not rigorous) solution to the (1,∞) -run-length limited (RLL) system on the hexagonal lattice. Furthermore, only exponential-time algorithms are known for the related problem of counting the exact number of constrained two-dimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a two-dimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graph-theoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique, we derive a closed-form solution to the capacity related to the Path-Cover constraint in a two-dimensional triangular array (the resulting calculated capacity is 0.72399217...). Path-Cover is a generalization of the well known one-dimensional (0.1) b-RLL constraint for which the capacity is known to be 0.69424....

KW - Capacity of constrained systems

KW - Capacity of twodimensional constrained systems

KW - Fisher-Kasteleyn-Temperley (FKT) method

KW - Holographic reductions

KW - Networks of relations

KW - Spectral distribution of Toeplitz matrices

UR - http://www.scopus.com/inward/record.url?scp=43749104730&partnerID=8YFLogxK

U2 - 10.1109/TIT.2008.920245

DO - 10.1109/TIT.2008.920245

M3 - Article

AN - SCOPUS:43749104730

SN - 0018-9448

VL - 54

SP - 2179

EP - 2195

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 5

ER -