gf(2) Find null basis vectors of M

M = 000000 000110 010100 101111 100101 111010

Augment M with identity (I) of same size

M = 000000 000001 000110 000010 010100 000100 101111 001000 100101 010000 111010 100000

M expressed in convenient bit depth

M = 00000000000000000000000000000001 00000000000000000000001100000010 00000000000000000000101000000100 00000000000000000001011110001000 00000000000000000001001010010000 00000000000000000001110100100000

Transform M into row reduced echelon form (rref)

M = 00000000000000000001000100111110 00000000000000000000100100000110 00000000000000000000010100011000 00000000000000000000001100000010 00000000000000000000000010101100 00000000000000000000000000000001

M in augmented form

M = 100010 111110 010010 000110 001010 011000 000110 000010 000001 101100 000000 000001

Highlight null basis vectors (where left side of row is zero)

M = 100010 111110 010010 000110 001010 011000 000110 000010 000001 101100 000000 000001

Null basis vectors

nv₀ = b00000000000000000000000000000001 nv₀ = 0x00000001 nv₀ = 1