On a Property of the Factorizing Codes

Felice, Clelia De

doi:10.1142/s0218196799000217

Cited by 8 publications

(2 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact we can point out that this arrangement has additional properties, through the notion of a good arrangement of a set. This is a slightly technical notion introduced in [13,15]. Roughly, it is an arrangement of its words in a matrix that can be recursively constructed thanks to the recursive constructions of the pairs (R h , T k ).…”

Section: Good Arrangementsmentioning

confidence: 99%

Finite maximal codes and factorizations of cyclic groups

Felice¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes, namely the factorization conjecture and the triangle conjecture, proposed by Perrin and Schützemberger. The latter concerns finite codes Y which are subsets of a * Ba * , where a is a letter and B is an alphabet not containing a. A structural property of finite maximal codes has recently been shown by Zhang and Shum. It exhibits a relationship between finite maximal codes and factorizations of cyclic groups. With the aim of highlighting the links between this result and other older ones on maximal and factorizing codes, we give a simpler and a new proof of this result. As a consequence, we prove that for any finite maximal code X ⊆ (B ∪ {a}) * containing the word a pq , where p, q are prime numbers, X ∩ a * Ba * satisfies the triangle conjecture. Let n be a positive integer that is a product of at most two prime numbers. We also prove that it is decidable whether a finite code Y ∪ a n ⊆ a * Ba * ∪ a * is included in a finite maximal code and that, if this holds, Y ∪ a n is included in a code that also satisfies the factorization conjecture.

show abstract

Section: Good Arrangementsmentioning

confidence: 99%