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.

翻译：我们重新审视了约束系统中经典的Gilbert-Varshamov（GV）界。1991年，Kolesnik和Krachkovsky指出GV界可通过求解某个优化问题来确定。随后，Marcus和Roth（1992）对该优化问题进行了修正，并在多种情形下改进了GV界。本文提供了求解这两个优化问题的显式数值方法，从而计算出相应的界。我们进一步证明，在绘制相应曲线时，这些方法可被简化。当图表示仅包含单一状态时，我们给出了两种界的显式公式。