# Some decompositions of Bernoulli sets and codes

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2005)

- Volume: 39, Issue: 1, page 161-174
- ISSN: 0988-3754

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topLuca, Aldo de. "Some decompositions of Bernoulli sets and codes." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 39.1 (2005): 161-174. <http://eudml.org/doc/245194>.

@article{Luca2005,

abstract = {A decomposition of a set $X$ of words over a $d$-letter alphabet $A=\lbrace a_1,\ldots ,a_d\rbrace $ is any sequence $X_1,\ldots , X_d, Y_1,\ldots , Y_d$ of subsets of $A^*$ such that the sets $X_i$, $i=1,\ldots , d,$ are pairwise disjoint, their union is $X$, and for all $i$, $1\le i\le d$, $X_i\sim a_iY_i$, where $\sim $ denotes the commutative equivalence relation. We introduce some suitable decompositions that we call good, admissible, and normal. A normal decomposition is admissible and an admissible decomposition is good. We prove that a set is commutatively prefix if and only if it has a normal decomposition. In particular, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition.},

author = {Luca, Aldo de},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {Bernoulli sets; codes; decompositions; commutative equivalence; Bernoulli distributions; prefix set},

language = {eng},

number = {1},

pages = {161-174},

publisher = {EDP-Sciences},

title = {Some decompositions of Bernoulli sets and codes},

url = {http://eudml.org/doc/245194},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Luca, Aldo de

TI - Some decompositions of Bernoulli sets and codes

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 1

SP - 161

EP - 174

AB - A decomposition of a set $X$ of words over a $d$-letter alphabet $A=\lbrace a_1,\ldots ,a_d\rbrace $ is any sequence $X_1,\ldots , X_d, Y_1,\ldots , Y_d$ of subsets of $A^*$ such that the sets $X_i$, $i=1,\ldots , d,$ are pairwise disjoint, their union is $X$, and for all $i$, $1\le i\le d$, $X_i\sim a_iY_i$, where $\sim $ denotes the commutative equivalence relation. We introduce some suitable decompositions that we call good, admissible, and normal. A normal decomposition is admissible and an admissible decomposition is good. We prove that a set is commutatively prefix if and only if it has a normal decomposition. In particular, we consider decompositions of Bernoulli sets and codes. We prove that there exist Bernoulli sets which have no good decomposition. Moreover, we show that the classical conjecture of commutative equivalence of finite maximal codes to prefix ones is equivalent to the statement that any finite and maximal code has an admissible decomposition.

LA - eng

KW - Bernoulli sets; codes; decompositions; commutative equivalence; Bernoulli distributions; prefix set

UR - http://eudml.org/doc/245194

ER -

## References

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