gf(2) Find null basis vectors of M

M = 00000000 00000110 00010100 01001000 01100001 10010101 01110110 01011000

Augment M with identity (I) of same size

M = 00000000 00000001 00000110 00000010 00010100 00000100 01001000 00001000 01100001 00010000 10010101 00100000 01110110 01000000 01011000 10000000

M expressed in convenient bit depth

M = 00000000000000000000000000000001 00000000000000000000110000000010 00000000000000000010100000000100 00000000000000001001000000001000 00000000000000001100001000010000 00000000000000010010101000100000 00000000000000001110110001000000 00000000000000001011000010000000

Transform M into row reduced echelon form (rref)

M = 00000000000000010000000011111110 00000000000000001001000000001000 00000000000000000101000011000010 00000000000000000010000010001000 00000000000000000000100010001100 00000000000000000000010010001110 00000000000000000000001011011010 00000000000000000000000000000001

M in augmented form

M = 10000000 11111110 01001000 00001000 00101000 11000010 00010000 10001000 00000100 10001100 00000010 10001110 00000001 11011010 00000000 00000001

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

M = 10000000 11111110 01001000 00001000 00101000 11000010 00010000 10001000 00000100 10001100 00000010 10001110 00000001 11011010 00000000 00000001

Null basis vectors

nv₀ = b00000000000000000000000000000001 nv₀ = 0x00000001 nv₀ = 1