  The zero-error channel capacity is the maximum asymptotic rate that can be
reached with error probability exactly zero, instead of a vanishing error
probability. The nature of this problem, essentially combinatorial rather than
probabilistic, has led to various researches both in Information Theory and
Combinatorics. However, the zero-error capacity is still an open problem, for
example the capacity of the noisy-typewriter channel with 7 letters is unknown.
In this article, we propose a new approach to construct optimal zero-error
codes, based on the concatenation of words of variable-length, taken from a
generator set. Three zero-error variable-length coding schemes, referred to as
"variable-length coding", "intermingled coding" and "automata-based coding",
are under study. We characterize their asymptotic performances via linear
difference equations, in terms of simple properties of the generator set, e.g.
the roots of the characteristic polynomial, the spectral radius of an adjacency
matrix, the inverse of the convergence radius of a generator series. For a
specific example, we construct an "intermingled" coding scheme that achieves
asymptotically the zero-error capacity.
