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Let’s revisit counting in binary. For this example, we will count from 0 to 7 with 3-digit binary numbers.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Decimal	Binary
0	`000`
1	`001`
2	`010`
3	`011`
4	`100`
5	`101`
6	`110`
7	`111`

In a mechanical counter, these 0s and 1s could be represented with switches or currents: off to represent “0”, and on to represent “1”. Thus, a number like 5 would be represented with 3 wires in the “on, off, on” positions respectively.

Let’s add 1 to the number 5 to get 6. The second and third wires need to be switched to give the new position of “on, on, off”. However, due to delays in switching, the second and third wires might not switch at the same time. This might give incorrect values of “on, off, off” (4) or “on, on, on” (7) if the wires are to be read at this time.

A solution to this is to come up with a new representation for numbers. We still want to use the binary digits, since each wire still only has 2 states, off and on. Let’s take a look at the following binary representation of the numbers from 0 to 7.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Decimal	Gray code
0	`000`
1	`001`
2	`011`
3	`010`
4	`110`
5	`111`
6	`101`
7	`100`

This is the Gray code representation, also called the reflected binary code. In this new representation, there’s only 1 bit of difference between any 2 consecutive numbers. This way, the value read will always be correct, even when the switches are changing. A nice bonus is that “100” (7) is also only 1 bit away from resetting back to “000” (0), forming a cycle.

But does this generalise to bit strings of other lengths? Yes! Let’s take a look at the construction. First, we start with the values 0 and 1, represented with the bits 0 and 1.

Gray Code

Prerequisites

Binary

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Binary

Gray Code