Density of Ham- and Lee- non-isometric k-ary Words

Isometric k-ary words have been defined referring to the Hamming and the Lee distances. A word is non-isometric if and only if it has a prefix at distance 2 from the suffix of same length; such a prefix is called 2-error overlap. The limit density of isometric binary words based on the Hamming distance has been evaluated by Klavzar and Shpectorov, obtaining that about 8% of all binary words are isometric. In this paper, the issue is addressed for k-ary words and referring to the Hamming and the Lee distances. Actually, the only meaningful case of Lee-isometric k-ary words is when k=4. It is proved that, when the length of words increases, the limit density of quaternary Ham-isometric words is around 17%, while the limit density of quaternary Lee-isometric words is even bigger, it is about 30%. The results are obtained using combinatorial methods and algorithms for counting the number of k-ary isometric words

Asymptotic number of isometric generalized Fibonacci cubes

AbstractFor a binary word f, let Qd(f) be the subgraph of the d-dimensional cube Qd induced on the set of all words that do not contain f as a factor. Let Gn be the set of words f of length n that are good in the sense that Qd(f) is isometric in Qd for all d. It is proved that limn→∞|Gn|/2n exists. Estimates show that the limit is close to 0.08, that is, about eight percent of all words are good