Gray code is permutation of $\{0,1,2,\dots,2^n-1\}$ such that each of consecutive number is differs only one bit in binary representation.
Example for $n = 3$
$000\\ 001\\ 011\\ 010\\ 110\\ 111\\ 101\\ 100$
Let $s_k$ is bit position of transition $k$ to $k+1$. In example for $n=3$ above are $s_1=3,s_2=2,s_3=3,s_4=1$ and so on
I define adjacent gray code is each consecutive number is differs in adjacent bit. It is, if $s_k=j$ than $s_{k+1}$ is $j-1$ or $j+1$
Example for $n=4$
$0000\\ 0001\\ 0011\\ 0111\\ 0101\\ 0100\\ 0110\\ 0010\\ 1010\\ 1110\\ 1100\\ 1101\\ 1111\\ 1011\\ 1001\\ 1000$
Can anyone design a good algorithm to look for adjacent gray code for $n$ large enough? Maybe it's acceptable for $n\leq10$