We introduce a novel framework for implementing error-correction in constrained systems. The main idea of our scheme, called Quantized-Constraint Concatenation (QCC), is to employ a process of embedding the codewords of an error-correcting code in a constrained system as a (noisy, irreversible) quantization process. This is in contrast to traditional methods, such as concatenation and reverse concatenation, where the encoding into the constrained system is reversible. The possible number of channel errors QCC is capable of correcting is linear in the block length $n$, improving upon the $O(\sqrt{n})$ possible with the state-of-the-art known schemes. For a given constrained system, the performance of QCC depends on a new fundamental parameter of the constrained system - its covering radius. Motivated by QCC, we study the covering radius of constrained systems in both combinatorial and probabilistic settings. We reveal an intriguing characterization of the covering radius of a constrained system using ergodic theory. We use this equivalent characterization in order to establish efficiently computable upper bounds on the covering radius.

翻译：我们提出了一种在约束系统中实现纠错的新框架。该方案的核心思想，称为量化约束级联（Quantized-Constraint Concatenation, QCC），是将纠错码的码字嵌入约束系统的过程视为一种（含噪、不可逆的）量化过程。这与传统的级联和逆级联方法形成对比，后者在约束系统中的编码是可逆的。QCC能够纠正的信道错误数量与块长度$n$呈线性关系，优于现有最优方案可实现的$O(\sqrt{n})$。对于给定的约束系统，QCC的性能取决于该约束系统的一个新基本参数——其覆盖半径。受QCC启发，我们在组合与概率两种设定下研究了约束系统的覆盖半径。我们利用遍历理论揭示了约束系统覆盖半径的一个有趣刻画，并借助这一等价刻画建立了覆盖半径的高效可计算上界。