7,4 Hamming Codes

Figure 0. 7,4 Hamming Code Venn Diagram

0. Note

This is the fifth assignment from my Masters Applied Digital Information Theory Course which has never been published anywhere and I, as the author and copyright holder, license this assignment customized CC-BY-SA where anyone can share, copy, republish, and sell on condition to state my name as the author and notify that the original and open version available here.

1. Introduction

The previous 4th assignment demonstrate 1 bit error detection using 3,4 parity codes. On this assignment will be demonstrating 1 bit error correction using 7,4 hamming codes. On 3,4 parity codes we group 3 bits per block and perform exclusive or on each blocks to get a bit called the parity code bit and add it into the 4th bit of the blocks. A different approach for the 7,4 hamming codes we first group 4 bits per block, and then obtain the 3 hamming bit codes from the 4 bits for each blocks and add them which makes each blocks contained 7 bits. Suppose there are 4 bits as follows:

b1,b2,b3,b4

To get the hamming bit codes we do the following calculation:

b5=b1⊕b2⊕b3, b6=b1⊕b2⊕b4, b7=b1⊕b3⊕b4

Those bits will be added to the block:

b1,b2,b3,b4,b5,b6,b7

Following Table 1 are the complete list:

Table 1. Coding Table

On the receiver side the hamming codes is also constructed using the received first 4 bits of each blocks, then compare them with the hamming codes produced on the receiver side. Since the hamming codes were produced using the above equation (each codes uses different elements) we can know on which bit the error occurs, and a correction is performed. For example if receiver’s b5, b6, b7 is different from transmitter’s then an error on b1, if b5, b6 is different then an error occur on b2, if b5,b7 is different then an error occur on b3, if b6, b7 then b4, if b5 only then b5, if b6 only then b6, if b7 only then b7. Following Table 2 are the complete syndromes: