Algebraic framework of Gray Codes

The classical algorithmic procedure of encoding and decoding are as follows:

Encoding

q-ry digits $d$ to gray digits $g$ conversion.

$g_{i} = \begin{cases} d_{i} , & \text{if} \mod\left(\displaystyle{\sum_{j=1}^{i-1}{g_{j}}},2\right)=0 \\ q-1-d_{i} , & \text{if} \mod\left(\displaystyle \sum_{j=1}^{i-1}{g_{j}},2\right)=1 \end{cases}$

and $g_{1} = d_{1}$.

Decoding

$d_{i} = \begin{cases} g_{i} , & \text{if} \mod\left(\displaystyle{\sum_{j=1}^{i-1}{g_{j}}},2\right)=0 \\ q-1-g_{i} , & \text{if} \mod\left(\displaystyle \sum_{j=1}^{i-1}{g_{j}},2\right)=1 \end{cases}$

and $d_{1} = g_{1}$.

Linear Algebraic Formulation (N.Rethnakar 2020)

Example of generator matrix $G$ for binary to gray mapping is given by,

\[G=\begin{pmatrix} 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ 1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & 1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & 1 & 1 & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & 1 & 1 & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & 1 & 1 \end{pmatrix}\]

The decoding matrix $B=G^{-1}$ is given by,

\[ B=\begin{pmatrix} 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ 1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ 1 & 1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ 1 & 1 & 1 & 1 & {\color{gray}0} & {\color{gray}0} \\ 1 & 1 & 1 & 1 & 1 & {\color{gray}0} \\ 1 & 1 & 1 & 1 & 1 & 1 \end{pmatrix}\]

Illustration of Binary Code

Generalized $q$-ry Gray Code

\[G=\begin{pmatrix} 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ q-1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & q-1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & q-1 & 1 & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & q-1 & 1 & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & q-1 & 1 \end{pmatrix} \equiv \begin{pmatrix} 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ -1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & -1 & 1 & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & -1 & 1 & {\color{gray}0} & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & -1 & 1 & {\color{gray}0} \\ {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & {\color{gray}0} & -1 & 1 \end{pmatrix}_{\mathbb{F}_{q}}\]

Gray Encoding as Differentiation

Encoding matrix operation act as a forward discrete differentiation operation in $\mathbb{F}_{q}$.