A trace partitioned Gray code forq-ary generalized Fibonacci strings

Abstract

We provide a trace partitioned Gray code for the set of q-ary strings avoiding a pattern constituted by k consecutive equal symbols. The definition of this Gray code is based on two different constructions, according to the parity of q. This result generalizes, and is based on, a Gray code for binary strings avoiding k consecutive 0's.

Gray code orders for $$q$$ q -ary words avoiding a given factor

“…This particular cases are discussed in Section 4.2, and we show that such factors f , f ≥ 2, are the only ones giving Gray codes for forbidden factors not inducing zero periodicity. In particular, the Gray code obtained for A n q (0 ℓ ) is one of those defined in [1] as a generalization of a Gray code in [14]. Finally, for forbidden factors f for which ≺ nor ⊳ does not produce Gray codes on A n q (f ), we give simple transformations of f , and eventually obtain Gray codes for A n q (f ) (in order other than ≺ or ⊳).…”

“…In particular, the Gray code obtained for A n q (0 ℓ ) is one of those defined in [1] as a generalization of a Gray code in [14]. Finally, for forbidden factors f for which ≺ nor ⊳ does not produce Gray codes on A n q (f ), we give simple transformations of f , and eventually obtain Gray codes for A n q (f ) (in order other than ≺ or ⊳).…”

“…If f ∈ U q and q is even, then f does not induce zero periodicity. Indeed, let for example f = b0 with b a suffix of (1(q − 1)0 m ) −∞ be as in relation (1). Then either the first word in b|A ∞ q (f ) when b has even parity, or the last word in b|A ∞ q (f ) when b has odd parity, has ultimate period 1(q − 1)0 m .…”

“…In Gray code context, Squire in his early paper [12] explores the possibility of listing factor avoiding words such that consecutive words differ in only one position, and by 1 or −1 in this position, and in [14] is given a Gray code and a generating algorithm for binary words avoiding ℓ consecutive 1s. The result in [14] was recently generalized in [1] where two Gray codes (one prefix partitioned and the other trace partitioned) for q-ary words avoiding a factor constituted by ℓ consecutive equal symbols are given.…”

Non-overlapping matrices

“…As a further development, it could be interesting to investigate about the possibility to list the matrices in a Gray code sense, following the studies started in [9], [12], [13], [14], [15], where different Gray codes are defined for several set of strings. Clearly, the variable dimension of the matrices has to be considered in a possible definition of a suitable Gray code.…”