gf(2) Find null basis vectors of M

M = 0000000 0000110 0010100 1001000 1010011 1101111 0111111

Augment M with identity (I) of same size

M = 0000000 0000001 0000110 0000010 0010100 0000100 1001000 0001000 1010011 0010000 1101111 0100000 0111111 1000000

M expressed in convenient bit depth

M = 00000000000000000000000000000001 00000000000000000000011000000010 00000000000000000001010000000100 00000000000000000100100000001000 00000000000000000101001100010000 00000000000000000110111100100000 00000000000000000011111101000000

Transform M into row reduced echelon form (rref)

M = 00000000000000000100000100010110 00000000000000000010000100101010 00000000000000000001000101110110 00000000000000000000100100011110 00000000000000000000010101110010 00000000000000000000001101110000 00000000000000000000000000000001

M in augmented form

M = 1000001 0010110 0100001 0101010 0010001 1110110 0001001 0011110 0000101 1110010 0000011 1110000 0000000 0000001

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

M = 1000001 0010110 0100001 0101010 0010001 1110110 0001001 0011110 0000101 1110010 0000011 1110000 0000000 0000001

Null basis vectors

nv₀ = b00000000000000000000000000000001 nv₀ = 0x00000001 nv₀ = 1