325 Hamming Code (7,4) :: Quicker, easier and cheaper to make your own chip!

How it works

This implementation of the (7,4) Hamming Code allows for the same input to be used for encoding and decoding, with dynamic selection of the mode using the MSB of the input.

Hamming Encoder (7,4) Overview

The Hamming (7,4) encoder is a linear error-correcting code that encodes 4 data bits into 7 bits by adding 3 parity bits, which can detect and correct a single-bit error.

Parity Format

{p1 p2 p3}

Data Format

{d1 d2 d3 d4}

Input

An 8-bit input "ui" with the following format (note the form is {7 6 5 4 3 2 1 0})

Input Pins

ui[0] - Bit 0 for 4-bit data input, d4
ui[1] - Bit 1 for 4-bit data input, d3
ui[2] - Bit 2 for 4-bit data input, d2
ui[3] - Bit 3 for 4-bit data input, d1
ui[4] - X
ui[5] - X
ui[6] - X
ui[7] - Mode Selector (0 => Encode, uses ui[3:0]; 1 => Decode, uses ui[6:0])

Output

An 8-bit output "uo" with the following format (note the form is {7 6 5 4 3 2 1 0})

Output Pins

uo[0] - Bit 0 for 7-bit encoded output, d4
uo[1] - Bit 1 for 7-bit encoded output, d3
uo[2] - Bit 2 for 7-bit encoded output, d2
uo[3] - Bit 3 for 7-bit encoded output, p3
uo[4] - Bit 4 for 7-bit encoded output, d1
uo[5] - Bit 5 for 7-bit encoded output, p2
uo[6] - Bit 6 for 7-bit encoded output, p1
uo[7] - X

Encode Mode

Encode Mode is selected by setting the MSB of the input (bit 7) LOW (0).
If encode mode is chosen, the encoder will use bits 3:0 as the four data bits to be encoded, and produce a 7-bit encoded output.
Bit 6:4 are not involved in any encoding.

Encode Mode Input Format

{selector, X, X, X, d1, d2, d3, d4}

Encode Mode Output Format

{p1, p2, d1, p3, d2, d3, d4}

Parity Bit Calculations

p1 covers bits d1, d2, and d4.
- p1 = d1 XOR d2 XOR d4
p2 covers bits d1, d3, and d4.
- p2 = d1 XOR d3 XOR d4
p3 covers bits d2, d3, and d4.
- p3 = d2 XOR d3 XOR d4

Expected Outputs of Encode Mode

0XXXd1d2d3d4 -> 0p1p2d1p3d2d3d4
00000000 -> 00000000
00000010 -> 00101010
00000001 -> 01101001
00000011 -> 01000011
00000100 -> 01001100
00000101 -> 00100101
00000110 -> 01100110
00000111 -> 00001111
00001000 -> 01110000
00001001 -> 00011001
00001010 -> 01011010
00001011 -> 00110011
00001100 -> 00111100
00001101 -> 01010101
00001110 -> 00010110
00001111 -> 01111111

Hamming Decoder (7,4) Overview

The decoder checks the received 7-bit word for errors and corrects a single-bit error if detected. The process involves recalculating the parity bits and comparing them with the received parity.

Decode Mode

Decode Mode is selected by setting the MSB of the input (bit 7) HIGH (1).
If decode mode is chosen, the decoder will use bits 7:0, both the data and parity bits, and produce a 7-bit decoded output. The decoded output will be the originally encoded input as long as there were less than 2 flipped bits between encoder output and decoder input.

Decode Mode Input Format

{p1, p2, d1, p3, d2, d3, d4}

Decode Mode Output Format

{p1, p2, d1, p3, d2, d3, d4}

a maximum of 1 bit could be flipped at position {S2, S1, S0}.

Syndrome Calculation

The syndrome indicates the position of an error (if any):

S0 is recalculated using the same bits used to calculate p1 during encoding:
- S0 = p1' XOR d1 XOR d2 XOR d4
S1 recalculates p2:
- S1 = p2' XOR d1 XOR d3 XOR d4
S2 recalculates p3:
- S2 = p3' XOR d2 XOR d3 XOR d4

Error Correction

The syndrome {S2, S1, S0} gives the error location:

If the syndrome is 000, no error is detected.
If the syndrome is non-zero, the position of the error corresponds to the syndrome value (1 for the least significant bit, 7 for the most significant bit).
E.g. if syndrome is 010, then. Our error bit is at bit 4
If an error is detected, flip the bit at the position indicated by the syndrome.

How to test

Testing can be done by applying known data inputs with LOW as the value of the 7th bit (encode mode), and ensuring that the output is the expected encoding value (see table of expected outputs in encode mode).

Similarly, known encoded values can by used as input, with the 7th bit as HIGH (decode mode), and we can ensure that the output is the exact same as the original encoded value, even if we flip 1 bit. This should be done for each of the 7 bits for all encoded values

External hardware

TBD based on implementation.

#	Input	Output
0	LSB/Bit 0 for 4-bit Encoder Input OR LSB/Bit 0 for 7-bit Decoder Input	LSB/Bit 0 for 7-bit Encoder OR Decoder Output
1	Bit 1 for 4-bit Encoder Input OR Bit 1 for 7-bit Decoder Input	Bit 1 for 7-bit Encoder OR Decoder Output
2	Bit 2 for 4-bit Encoder Input OR Bit 2 for 7-bit Decoder Input	Bit 2 for 7-bit Encoder OR Decoder Output
3	MSB/Bit 3 for 4-bit Encoder Input OR Bit 3 for 7-bit Decoder Input	Bit 3 for 7-bit Encoder OR Decoder Output
4	Bit 4 for 7-bit Decoder Input	Bit 4 for 7-bit Encoder OR Decoder Output
5	Bit 5 for 7-bit Decoder Input	Bit 5 for 7-bit Encoder OR Decoder Output
6	MSB/Bit 6 for 7-bit Decoder Input	MSB/Bit 6 for 7-bit Encoder OR Decoder Output
7	Mode Selector (0 => Encode, uses ui[3:0]; 1 => Decode, uses ui[6:0])	Mode Selector (0 => Encode; 1 => Decode)

325 Hamming Code (7,4)