On locally balanced Gray codes

We consider locally balanced Gray codes. We say that a Gray code is locally balanced if each “short” subword of transition sequence contains all letters of the set $\{1,2,\dots,n\}$. The minimal length of such subwords is called the window width of the code. We show that for each $n\ge3$ there exists a Gray code with window width not greater than $n+3\lfloor\log n\rfloor$. Tab. 3, bibliogr. 10.