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. The study of finite-type constrained codes, for which the set of forbidden substrings is finite, dated back to a pioneering and influential work of Shannon in the 1940s. To the best of our knowledge, for roughly 80 years, people have known essentially only one method, i.e., the “spectral method”, that computes the rate and the cardinality of finite-type constrained codes. We show that there is a surprisingly powerful method arising from enumerative combinatorics, which applies the Goulden-Jackson cluster theorem (previously not known to the coding community), that serves as an alternative method to compute the code rate and the exact cardinality of each fixed length, of these codes. Moreover, the computation can be done by solving a system of linear equations of size equal to the number of constraints, and the time complexity improves that of the spectral method when the number of constraints is relatively small. More interestingly, our new method has the flexibility that it also applies to constrained codes defined by an infinite number of forbidden substrings. As an example, variable-length non-overlapping codes have potential applications in DNA storage. Bilotta, also Wang and Wang asked for an explicit upper bound on the maximum cardinality of these codes. By applying the cluster method in concert with other tools in analytic combinatorics, we obtain such a bound, thereby giving an affirmative answer to their question. Moreover, our bound is tight when the code length divides the alphabet size, as shown by a construction of Blackburn. Lastly, we 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 former and the convergence radius of the generating function used in the latter.

Tanner codes are error-correcting codes built from a bipartite graphGand a short inner codeC0. Expander codes are a special type of Tanner code, where the graph is highly interconnected, ensuring stronger error correction capabilities. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that δ ∈ [0, 1] andd0∈ N must satisfy, so thateverybipartite expanderGwith vertex expansion ratio δ andeverylinear inner codeC0with minimum distanced0together define an expander code that corrects Ω(n) errors inO(n) time? ForC0being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT’96) showed that δ > 3/4 is sufficient; later, Viderman (ACM-TOCT’13) improved this to δ > 2/3 - Ω(1) and he also showed that δ > 1/2 is necessary. For general linear codeC0, the previously best-known result of Dowling and Gao (IEEE-TIT’18) showed thatd0= Ω(cδ-2) is sufficient, wherecis the left-degree ofG. We present a near-optimal solution to the above problem for generalC0by showing that δd0> 3 is sufficient and δd0> 1 is necessary, thereby significantly improving Dowling-Gao’s result. To prove the sufficient condition, we present two novel algorithms for decoding arbitrary expander codes withδd0> 3, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius. To prove the necessary condition, we generalize the aforementioned necessary result of Viderman, and construct for every pair of δ,d0with δd0=1, an expander code with constant distance, that only corrects a constant number of errors.

Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph $G$ together with a linear inner code $C_0$. Expander codes are Tanner codes whose defining bipartite graph $G$ has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that $δ$ and $d_0$ must satisfy, so that \textit{every} bipartite expander $G$ with vertex expansion ratio $δ$ and \textit{every} linear inner code $C_0$ with minimum distance $d_0$ together define an expander code that corrects $Ω(n)$ errors in $O(n)$ time? For $C_0$ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that $δ>3/4$ is sufficient; later Viderman (ACM-TOCT'13) improved this to $δ>2/3-Ω(1)$ and he also showed that $δ>1/2$ is necessary. For general linear code $C_0$, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that $d_0=Ω(cδ^{-2})$ is sufficient, where $c$ is the left-degree of $G$. In this paper, we give a near-optimal solution to the above question for general $C_0$ by showing that $δd_0>3$ is sufficient and $δd_0>1$ is necessary, thereby also significantly improving Dowling-Gao's result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.