Examples

Recursive construction

Recursive procedure

Reflect $C[n-1]$, shift by $q^{n-1}$ and augment (TBD).

For $n \in \mathbb{N}$, positive integer and $N = 2^{n}$. A Gray code $G_{n}$ is an tuple $G_{n} = (X_{1},X_{2},...,X_{N})$ which satisfies the following properties:

$X_{1}, X_{2}, ... , X_{N}$ are binary sequences (of length $n$) corresponding to the binary representation of the numbers $0, 1, \ldots , N − 1$, arranged in a specific order,
For any $0 \le j \le N-1$, adjacent pairs $X_{j},X_{j+1}$ differ in only one position (i.e., only one of the $n$ bits would differ),
The start and end sequences (i.e., sequences $X_{1}$ and $X_{N}$ differ in just one position.

Gray code $G_{n}$ can be recursively constructed as follows. Start with $G_{1} = (0,1)$ and for $N=2^n, n \ge 1$, Let $G_{n} = \left(X_{1},\ldots,X_{N−1},X_{N}\right)$,

\[G_{n+1} = \left(0X_{1},\ldots,0X_{N−1},0X_{N},1X_{N},1X_{N−1},...,1X_{1}\right).\]

Illustration

using Plots
function plotmatrix(A;kwargs...)
    a,b=size(A)
    X = transpose(repeat(1:b, 1, a))[:]
    Y = repeat(a:-1:1, b)[:]
	scatter(X,Y, marker_z = A[:], marker=:rect,markersize = 4,  color = :viridis,aspectratio=1,ylims=[0,size(G,1)+1],alpha=1,label=:none,colorkey=:none,axis=:none;kwargs...)

julia> plotmatrix(gray(6));
julia> plotmatrix(G,size=(800,400),color=:summer)
julia> plotmatrix(G,size=(800,200),color=:summer,markersize=7,xlims=[1,size(G,2)+0],ylims=[1/2,size(G,1)-0])
end

Binary Gray Code $n=4$

Binary Gray Code $n=5$

Binary Gray Code $n=6$

Linear Algebraic method

TBD $g=Gb$ and $b=Bg$, where $G$ is a Jordan matrix, which is

julia> n,q=4,2
julia> GrayCoding.GrayMatrix(n,q)
4×4 Matrix{Int64}:
 1  0  0  0
 1  1  0  0
 1  1  1  0
 1  1  1  1
4×4 Matrix{Int64}:
 1  0  0  0
 1  1  0  0
 0  1  1  0
 0  0  1  1
4×16 Matrix{Int64}:
 0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1
 0  0  0  0  1  1  1  1  0  0  0  0  1  1  1  1
 0  0  1  1  0  0  1  1  0  0  1  1  0  0  1  1
 0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1
4×16 Matrix{Int64}:
 0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1
 0  0  0  0  1  1  1  1  1  1  1  1  0  0  0  0
 0  0  1  1  1  1  0  0  0  0  1  1  1  1  0  0
 0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0

julia> G,B,g,b=GrayCoding.GrayMatrix(10,5);
julia> G
10×10 Matrix{Int64}:
 1  0  0  0  0  0  0  0  0  0
 1  1  0  0  0  0  0  0  0  0
 1  1  1  0  0  0  0  0  0  0
 1  1  1  1  0  0  0  0  0  0
 1  1  1  1  1  0  0  0  0  0
 1  1  1  1  1  1  0  0  0  0
 1  1  1  1  1  1  1  0  0  0
 1  1  1  1  1  1  1  1  0  0
 1  1  1  1  1  1  1  1  1  0
 1  1  1  1  1  1  1  1  1  1
 julia>B
 10×10 Matrix{Int64}:
 1  0  0  0  0  0  0  0  0  0
 4  1  0  0  0  0  0  0  0  0
 0  4  1  0  0  0  0  0  0  0
 0  0  4  1  0  0  0  0  0  0
 0  0  0  4  1  0  0  0  0  0
 0  0  0  0  4  1  0  0  0  0
 0  0  0  0  0  4  1  0  0  0
 0  0  0  0  0  0  4  1  0  0
 0  0  0  0  0  0  0  4  1  0
 0  0  0  0  0  0  0  0  4  1