Evaluating the Gilbert-Varshamov Bound for Constrained Systems
Keshav Goyal, Han Mao Kiah
TL;DR
The paper revisits the Gilbert-Varshamov bound for constrained systems and develops explicit numerical procedures to compute the GV bound and the GV-MR improvement by solving convex optimization problems via a blend of modified power iteration and Newton-Raphson methods. It derives dual formulations that reduce to dominant eigenvalue problems of reduced distance matrices, enabling both general algorithms and compact single-state expressions for efficient GV-curve plotting. Through SWCC, RLL, and SECC examples, the approach yields tight GV-type bounds and demonstrates significant improvements over prior bounds, even for large state spaces. The methods offer scalable tools for evaluating asymptotic code rates in constrained settings with potential impact on data storage and communications where constraints are essential. The work also provides fast, curve-based plotting techniques that avoid heavy Newton iterations when mapping the GV curve.
Abstract
We revisit the well-known Gilbert-Varshamov (GV) bound for constrained systems. In 1991, Kolesnik and Krachkovsky showed that GV bound can be determined via the solution of some optimization problem. Later, Marcus and Roth (1992) modified the optimization problem and improved the GV bound in many instances. In this work, we provide explicit numerical procedures to solve these two optimization problems and hence, compute the bounds. We then show the procedures can be further simplified when we plot the respective curves. In the case where the graph presentation comprise a single state, we provide explicit formulas for both bounds.