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Find binary solution over non square matrix

asked 3 years ago

Sanu gravatar image

I have system of m homogeneous linear equations of n variables over GF(2) where m>n. We know all zero is a solution. How to find any other solution? I know there are other solutions as rank of the corresponding matrix < n.

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answered 3 years ago

Max Alekseyev gravatar image

updated 3 years ago

So you have a matrix equation over GF(2) of the form Mx=0,

where M is an m×n matrix. The solutions to this equation form the (right) kernel of M, which is a linear subspace of GF(2)n.

In Sage, the kernel can be computed with function .right_kernel() - like in the example below:

sage: M = matrix(GF(2), [[1, 0, 1, 1], [1, 0, 0, 1]])                                                                                                                                                      
sage: K = M.right_kernel()                                                                                                                                                                                 
sage: K                                                                                                                                                                                                    
Vector space of degree 4 and dimension 2 over Finite Field of size 2
Basis matrix:
[1 0 0 1]
[0 1 0 0]
sage: list(K)                                                                                                                                                                                              
[(0, 0, 0, 0), (1, 0, 0, 1), (0, 1, 0, 0), (1, 1, 0, 1)]

In this example, the kernel K of M is spanned by two vectors. So, K is composed of 22=4 vectors, including the zero vector.

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Last updated: Jul 06 '21