Constrained coding upper bounds via Goulden-Jackson cluster theorem
Yuanting Shen, Chong Shangguan, Zhicong Lin, Gennian Ge
TL;DR
This work develops a unified, exact framework for constrained coding by employing the Goulden-Jackson cluster theorem to finite-type constraints defined by a finite forbidden set $F$. It yields a closed-form generating function $f_F(x)=\frac{T(x)}{S(x)}$ for the exact code-cardinality sequence $\{N_F(n)\}$ and proves that the capacity satisfies $\mathrm{Cap}(F)=\log_q\frac{1}{x_0}$ where $x_0$ is the smallest positive root of $S(x)$; further, the radius $R$ equals $1/\lambda(G_F)$, linking to the spectral method. The results provide a constructive procedure to compute $S$ and $T$ in $O(|F|^3)$ and to obtain recurrences for $N_F(n)$, while also enabling explicit upper bounds for infinite-forbidden-substring codes, such as variable-length non-overlapping codes. The paper demonstrates the method on several finite-type codes (e.g., palindrome-avoiding and least-periodicity-avoiding constraints) and a multi-constraint example, deriving accurate capacities and exact $N_F(n)$ recurrences, thereby offering practical tools for DNA storage and related constrained-coding applications. Overall, the cluster approach complements and illuminates the spectral method, providing an exact, scalable pathway to both capacities and exact code sizes across a broad class of constrained codes.
Abstract
Motivated by applications in DNA-based data storage, constrained codes have attracted a considerable amount of attention from both academia and industry. We study the maximum cardinality of constrained codes for which the constraints can be characterized by a set of forbidden substrings, where by a substring we mean some consecutive coordinates in a string. For finite-type constrained codes (for which the set of forbidden substrings is finite), one can compute their capacity (code rate) by the ``spectral method'', i.e., by applying the Perron-Frobenious theorem to the de Brujin graph defined by the code. However, there was no systematic method to compute the exact cardinality of these codes. We show that there is a surprisingly powerful method arising from enumerative combinatorics, which is based on the Goulden-Jackson cluster theorem (previously not known to the coding community), that can be used to compute not only the capacity, but also the exact formula for the cardinality of these codes, for each fixed code length. Moreover, this can be done by solving a system of linear equations of size equal to the number of constraints. We also show that the spectral method and the cluster method are inherently related by establishing a direct connection between the spectral radius of the de Brujin graph used in the first method and the convergence radius of the generating function used in the second method. Lastly, to demonstrate the flexibility of the new method, we use it to give an explicit upper bound on the maximum cardinality of variable-length non-overlapping codes, which are a class of constrained codes defined by an infinite number of forbidden substrings.