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Construction of a family of finite maximal codes. (English) Zbl 0667.68079
The author introduces a family $${\mathcal F}$$ of finite maximal codes over a two-letters alphabet $$A=\{a,b\}$$, defined as the set of finite codes C verifying: $C-1=p(A-1)S\quad \Leftrightarrow \quad A^*=SC^*P$ where P,S are finite subsets of $$A^*$$ such that either P or S is contained in $$a^*$$. It is shown that every finite maximal code with two b’s is in $${\mathcal F}$$. The main result of the paper consists on an algorithmical characterization of $${\mathcal F}$$. Indeed, an algorithm based on the effective resolution of the inequality: $a^ M=\sum_{m\in M}(M,m)a^ m\leq a^{M+I+1}+a^ I$ where I is a given subset of $${\mathbb{N}}$$ and M is a unknown subset of $${\mathbb{N}}$$ is proposed in order to compute any code of $${\mathcal F}$$. The paper ends with an application to the construction of all factorizations (T,R) of $${\mathbb{Z}}/n{\mathbb{Z}}$$- i.e., of subsets of $${\mathbb{N}}$$ such that each element of $$\{$$ 0,...,n-1$$\}$$ can be written uniquely as a sum modulo n of an element of T and of an element of R - such that $$a^ T$$ or $$a^ R$$ is one factor of a polynomial factorization of $$P(n)=1+a+...+a^{n-1}$$.
Reviewer: D.Krob

##### MSC:
 68Q45 Formal languages and automata 94A45 Prefix, length-variable, comma-free codes
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##### References:
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