Bifix codes and Sturmian words

We prove new results concerning the relation between bifix codes, episturmian words and subgroups offree groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set $F$ of words ($F$-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let $F$ be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an $F$-maximal bifix code of degree $d$ behaves just as the set of words of $F$ of length $d$. An $F$-maximal bifix code of degree $d$ in a Sturmian set of words on an alphabet with $k$ letters has $(k-1)d+1$ elements. This generalizes the fact that a Sturmian set contains $(k-1)d+1$ words of length $d$. Moreover, given an infinite word $x$, if there is a finite maximal bifix code $X$ of degree $d$ such that $x$ has at most $d$ factors of length $d$ in $X$, then $x$ is ultimately periodic. Our main result states that any $F$-maximal bifix code of degree $d$ on the alphabet $A$ is the basis of a subgroup of index $d$ of the free group on~$A$.

💡 Research Summary

The paper investigates the interplay between bifix codes, Sturmian (more precisely, strict episturmian) word sets, and subgroups of free groups. A bifix code is a set of words that is simultaneously prefix‑free and suffix‑free. Classical theory tells us that a maximal bifix code over an alphabet $A$ is a basis of a subgroup of the free group $A^{*}$ whose index equals the code’s degree $d$. The authors extend this framework to a more restrictive environment: a recurrent factorial set $F$ of words. They define an $F$‑maximal bifix code as a bifix code contained in $F$ that cannot be enlarged while staying inside $F$.

The central object of study is the case where $F$ is a Sturmian set, i.e., the set of all factors of a strict episturmian infinite word. Sturmian sets are characterized by having exactly $(k-1)n+1$ distinct factors of length $n$ when the alphabet has $k$ letters. The authors prove that any $F$‑maximal bifix code $X$ of degree $d$ in such a set contains precisely $(k-1)d+1$ words. This mirrors the classical factor count and shows that $X$ behaves exactly like the set of all length‑$d$ factors of $F$.

A major contribution is the theorem that every $F$‑maximal bifix code of degree $d$ is a basis of a subgroup of index $d$ in the free group on $A$. The proof proceeds by constructing the Rauzy graph of $F$, interpreting the code as a set of cycles that generate the graph’s fundamental group, and then showing that the subgroup generated by $X$ has the required index. Consequently, the combinatorial maximality inside $F$ translates directly into a group‑theoretic maximality in $A^{*}$.

Beyond the structural results, the paper derives a dynamical consequence: if an infinite word $x$ admits a finite maximal bifix code $X$ of degree $d$ such that $x$ contains at most $d$ distinct factors of length $d$ belonging to $X$, then $x$ must be ultimately periodic. The argument uses the fact that the subgroup generated by $X$ has finite index; a word that meets only a bounded number of $X$‑factors cannot explore infinitely many cosets, forcing it into a periodic orbit.

The authors also discuss how many classical properties of ordinary maximal bifix codes survive in the $F$‑maximal setting: the closure under taking inverses, the correspondence with return words, and the relationship with the factor complexity of $F$. They provide several examples illustrating the theory, including explicit constructions for binary and ternary Sturmian sets.

In the concluding section, the paper points out several avenues for future work. One direction is to generalize the results to broader classes of recurrent sets, such as Arnoux–Rauzy sequences or more general episturmian families. Another is to develop algorithmic procedures for computing $F$‑maximal bifix codes and for testing the periodicity condition efficiently. The authors suggest that the bridge they built between combinatorics on words, symbolic dynamics, and free group theory could be exploited in coding theory, data compression, and the study of low‑complexity dynamical systems. Overall, the work offers a deep and unified perspective on how maximal bifix codes encode both combinatorial and algebraic information within Sturmian environments.