How do we calculate the branch number of any given binary matrix?

How do we calculate the hamming weight, branch number of any nxn binary matrix defined in GF(2^n)?

Answer 1

[Srivardhan]

Hi Smita,

As per my understanding, you want to calculate the branch number and hamming weight of any nxn binary matrix.

In general, Hamming weight of binary matrix is the count of non-zero elements in the matrix. To calculate the branch number of given matrix we need to find absolute maximum in Walsh transform of the matrix. Subtract the maximum absolute value from 2^(n-1) to obtain the branch number.

We can use suitable algorithm to find Walsh transform of the matrix such as Fast Walsh Hadamard transform (FWHT).

Here is the sample code for reference:

function weight = hamming_weight(matrix)
    weight = sum(matrix(:));
end

function bn = branch_number(matrix)
    n = size(matrix, 1);
    transform = fwht(matrix);  % Fast Walsh–Hadamard transform
    abs_transform = abs(transform);
    max_value = max(abs_transform(:));
    bn = 2^(n-1) - floor(max_value);
end

% Example usage
matrix = [1 0 1; 0 1 0; 1 1 1];
weight = hamming_weight(matrix);
bn = branch_number(matrix);
disp(weight); 
disp(bn);

For further reference, please check out the links to learn more about hamming weight and branch number.

• https://www.mathworks.com/help/signal/ref/fwht.html
• number of 1 bits - hamming weight - set bits (tutorialcup.com)

I hope this helps!

[曦]

Can you explain why it is possible to calculate the branch number in this way, or do you have any relevant references.Thx