Improved Lower Bounds on Capacities of Symmetric 2-Dimensional Constraints using Rayleigh Quotients

A method for computing lower bounds on capacities of 2-dimensional constraints having a symmetric presentation in either the horizontal or the vertical direction is presented. The method is a generalization of the method of Calkin and Wilf (SIAM J. Discrete Math., 1998). Previous best lower bounds on capacities of certain constraints are improved using the method. It is also shown how this method, as well as their method for computing upper bounds on the capacity, can be applied to constraints which are not of finite-type. Additionally, capacities of 2 families of multi-dimensional constraints are given exactly.

💡 Research Summary

The paper presents a novel framework for deriving improved lower bounds on the capacities of two‑dimensional (2‑D) constrained systems that possess a symmetric presentation in either the horizontal or vertical direction. Capacity, defined as the exponential growth rate of the number of admissible arrays, is a fundamental metric in information theory, coding, and storage. Prior work by Calkin and Wilf (1998) introduced a Rayleigh‑quotient‑based method for a limited class of symmetric constraints, but its applicability was narrow and the resulting bounds were relatively modest.

The authors first formalize the notion of a “symmetric presentation.” A constraint has a symmetric presentation if its defining local rules are invariant under reflection in one coordinate direction (horizontal or vertical). This symmetry allows the transition matrix that encodes admissible row‑to‑row (or column‑to‑column) extensions to be expressed in a block‑diagonal or otherwise highly structured form. By exploiting this structure, the Rayleigh quotient
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