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L codes and number systems. (English) Zbl 0531.68027
Given a morphism $$h: V^*\to V^*$$ ($$V$$ is an alphabet), the authors define a mapping $$h: V^*\to V^*$$, by putting: $$\bar h(v_ 1\cdots v_ n)=h(v_ 1)h^ 2(v_ 2)\cdots h^ n(v_ n)$$, $$\bar h(\Lambda)=\Lambda$$. Now, $$h$$ is called an L-code (resp. almost L-code) if $$\bar h(a)=\bar h(b)$$ entails $$a=b$$, for all $$a,b\in V^*$$ (resp. for all $$a,b\in V^*$$, such that $$| a| \neq t$$ and $$| b| \neq t$$, where $$t$$ is a positive integer depending on $$h$$). As every code, i.e. injective morphism, is also an L-code (theorem 1), the concept of L-code appears to be a natural generalization of that of code. The authors give a nice characterization of L-codes in terms of so-called number systems, which allows to establish several fundamental properties of L-codes. The main attention is focused on unary L-codes, i.e. such $$h$$ that $$h(v)=v^ n$$ for $$v\in V$$ $$(n$$ depends on $$v)$$, and the central result (theorem 5) is: every morphism $$h$$, such that $$h(v)=v^ n$$, $$h(v')=v^ r$$ $$(v=\{v,v'\})$$, $$n\geq 2$$, $$r\geq 1$$, $$r\neq n$$, is either an L-code or an almost L-code. This fails to hold true for the three-element $$v$$, and to find analogous theorems for greater alphabets has been left an open problem.
The paper contains a number of closely related results. As the unary morphisms are usually non-codes, the afore-mentioned theorem shows indeed that the author’s new concept is a powerful one. The authors indicate cryptography as a possible domain of applications.
Reviewer: W. Buszkowski

MSC:
 68Q45 Formal languages and automata 68Q70 Algebraic theory of languages and automata 20M35 Semigroups in automata theory, linguistics, etc.
Keywords:
morphism; L-code; cryptography; number systems
Full Text:
References:
 [1] Kahn, D., The codebreakers, (1967), MacMillan New York [2] Lallement, G., Semigroups and combinatorial applications, (1979), Wiley New York · Zbl 0421.20025 [3] Rozenberg, G.; Salomaa, A., The mathematical theory of L system, (1980), Academic Press New York · Zbl 0365.68072 [4] Ryska, N.; Herda, S., Kryptographische verfahren in der datenverarbeitung, (1980), Springer Berlin · Zbl 0426.68107 [5] Salomaa, A., Formal languages, (1973), Academic Press New York · Zbl 0262.68025 [6] Salomaa, A., Jewels of formal language theory, (1981), Computer Science Press Rockville, MD · Zbl 0487.68063
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