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Gray Code (Non-weighted BCD)

Gray Code: The reflected binary code or gray code is an ordering of binary number systems where two successive values differ in only one bit.

Properties of Gray Code:

i) Gray code is non-weighted BCD code.

ii) It is a cyclic code, i.e. unit distance code (successive code words differ in one-bit position only).

iii) It is reflective code (Gray codes for all decimal numbers can be generated through the reflective property).

# Distance means if two codes have changed only 1 bit, then the distance between two codes is 1.

So, the beauty of gray code is if we choose any two successive numbers, the distance between two numbers is always 1 (unit) that’s why gray code also called unit distance code.

Binary to Gray code conversion:

Say, an 8-bit binary number is

The equivalent Gray code will be 

Where, 

Gray code to Binary conversion:

Say, an 8-bit Gray Code is

The equivalent binary will be

Where,

Reflective Code:

Gray code also called reflective code because 'n' least significant bits for ${\mathbf{2}}^{\mathbf{n}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}}$ through ${\mathbf{2}}^{\mathbf{n}\mathbf{ }}\mathbf{-}\mathbf{ }\mathbf{1}$ are the mirror images of those for 0 through ${\mathbf{2}}^{\mathbf{n}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}$.

Example: We take examples to understand how gray code is reflective. 

If see the above definition of reflective code that is 'n' least significant bits for ${\mathbf{2}}^{\mathbf{n}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}}$ through ${\mathbf{2}}^{\mathbf{n}\mathbf{ }}\mathbf{-}\mathbf{ }\mathbf{1}$are the mirror images of those for 0 through ${\mathbf{2}}^{\mathbf{n}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}$.

So, 

Here in this above example n = 4 (4 - bit binary number), so, ${\mathbf{2}}^{\mathbf{n}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}}$ to ${\mathbf{2}}^{\mathbf{n}\mathbf{ }}\mathbf{-}\mathbf{ }\mathbf{1}$ means ${\mathbf{2}}^{\mathbf{3}}$ to ${\mathbf{2}}^{\mathbf{4}\mathbf{ }}\mathbf{-}\mathbf{ }\mathbf{1}$ i.e. 8 to 15 and 0 to ${\mathbf{2}}^{\mathbf{n}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}$ is 0 to ${\mathbf{2}}^{\mathbf{3}\mathbf{ }}\mathbf{-}\mathbf{ }\mathbf{1}$ i.e. 0 to 7.

So, all 4th least significant bits of  8 to 15 are the mirror images of those codes for 0 to 7. 

When considering 3 bit, then n = 3,

So, all 3rd least significant bits of 4 to 8 are the mirror images of those codes for 0 to 3.