Constrained coding is a fundamental field in coding theory that tackles efficient communication through constrained channels. While channels with fixed constraints have a general optimal solution, there is increasing demand for parametric constraints that are dependent on the message length. Several works have tackled such parametric constraints through iterative algorithms, yet they require complex constructions specific to each constraint to guarantee convergence through monotonic progression. In this paper, we propose a universal framework for tackling any parametric constrained-channel problem through a novel simple iterative algorithm. By reducing an execution of this iterative algorithm to an acyclic graph traversal, we prove a surprising result that guarantees convergence with efficient average time complexity even without requiring any monotonic progression. We demonstrate the effectiveness of this universal framework by applying it to a variety of both local and global channel constraints. We begin by exploring the local constraints involving illegal substrings of variable length, where the universal construction essentially iteratively replaces forbidden windows. We apply this local algorithm to the minimal periodicity, minimal Hamming weight, local almost-balanced Hamming weight and the previously-unsolved minimal palindrome constraints. We then continue by exploring global constraints, and demonstrate the effectiveness of the proposed construction on the repeat-free encoding, reverse-complement encoding, and the open problem of global almost-balanced encoding. For reverse-complement, we also tackle a previously-unsolved version of the constraint that addresses overlapping windows. Overall, the proposed framework generates state-of-the-art constructions with significant ease while also enabling the simultaneous integration of multiple constraints for the first time.

翻译：约束编码是编码理论中处理通过受限信道高效通信的基础领域。尽管固定约束信道存在通用最优解，但近年来对依赖于消息长度的参数约束需求日益增长。已有研究通过迭代算法解决此类参数约束问题，但需要针对每种约束设计复杂的特定构造，才能通过单调递进保证收敛。本文提出一种通用框架，通过新颖的简单迭代算法解决任意参数约束信道问题。通过将该迭代算法的执行过程简化为无环图遍历，我们证明了一个令人惊讶的结果：即使无需任何单调递进，该算法仍能保证收敛且具有高效的平均时间复杂度。我们通过将该通用框架应用于多种局部与全局信道约束，验证了其有效性。首先探索涉及可变长度非法子串的局部约束，其中通用构造本质上通过迭代替换禁用窗口实现。我们将该局部算法应用于最小周期性约束、最小汉明重量约束、局部近似平衡汉明重量约束以及此前未解决的最小回文约束。继而探索全局约束，并在无重复编码、反向互补编码以及全局近似平衡编码这一开放问题上验证了所提构造的有效性。针对反向互补编码，我们还解决了此前未解决的涉及重叠窗口的约束版本。总体而言，该框架以显著简洁性生成最优构造，同时首次实现多约束的同步集成。