Goyal Keshav, Kiah Han Mao
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637121, Singapore.
Entropy (Basel). 2024 Apr 19;26(4):346. doi: 10.3390/e26040346.
We revisit the well-known Gilbert-Varshamov (GV) bound for constrained systems. In 1991, Kolesnik and Krachkovsky showed that the GV bound can be determined via the solution of an optimization problem. Later, in 1992, Marcus and Roth 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 that the procedures can be further simplified when we plot the respective curves. In the case where the graph presentation comprises a single state, we provide explicit formulas for both bounds.
我们重新审视了约束系统中著名的吉尔伯特 - 瓦尔沙莫夫（GV）界。1991年，科列斯尼克和克拉奇科夫斯基表明，可以通过一个优化问题的解来确定GV界。后来，在1992年，马库斯和罗斯修改了该优化问题，并在许多情况下改进了GV界。在这项工作中，我们提供了明确的数值程序来解决这两个优化问题，从而计算出界。然后我们表明，当绘制各自的曲线时，这些程序可以进一步简化。在图形表示仅包含单个状态的情况下，我们给出了两个界的明确公式。
