Hamming Weight

We can calculate the Minimum Distance# of a linear block code by finding the minimum Hamming Weight# of any non-zero code word in the code. And from here, we can know the error detection capability of a linear block code, \(\varepsilon\) with the equation:

To get the codewords from the equation, we need to first calculate out the total amount of possible messages using \(b^k\) where \(b\) is the base used for encoding. Then, fill out all the codewords based on \(I_k\) that aligned to the messages pattern, and we will get the codeword in the form \(P_i I_i\) where \(i\) is the row in \(G\). To find out others, we utilise the bit-wise XOR on \(P\) to get the desire message \(m_i\) based on \(I_i\), where the result is still in the form of \(P_i I_i\). Note that \(m_i\) and \(I_i\) should have same bit pattern. From here, we can calculate the Hamming Weight#.