gf(2) Find null basis vectors of M

M = 0000000000 0000000110 0000010100 0001001000 0100010000 1111111101 0010001110 1110111000 1001011010 0011010010

Augment M with identity (I) of same size

M = 0000000000 0000000001 0000000110 0000000010 0000010100 0000000100 0001001000 0000001000 0100010000 0000010000 1111111101 0000100000 0010001110 0001000000 1110111000 0010000000 1001011010 0100000000 0011010010 1000000000

M expressed in convenient bit depth

M = 00000000000000000000000000000001 00000000000000000011000000000010 00000000000000001010000000000100 00000000000000100100000000001000 00000000000010001000000000010000 00000000000111111110100000100000 00000000000001000111000001000000 00000000000111011100000010000000 00000000000100101101000100000000 00000000000001101001001000000000

Transform M into row reduced echelon form (rref)

M = 00000000000100000000000100001110 00000000000010000001000000010110 00000000000001000001100011101000 00000000000000100001100010100010 00000000000000010000000111011100 00000000000000001001000000000110 00000000000000000101100010101010 00000000000000000011000000000010 00000000000000000000001001001100 00000000000000000000000000000001

M in augmented form

M = 1000000000 0100001110 0100000010 0000010110 0010000011 0011101000 0001000011 0010100010 0000100000 0111011100 0000010010 0000000110 0000001011 0010101010 0000000110 0000000010 0000000000 1001001100 0000000000 0000000001

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

Null basis vectors

nv₀ = b00000000000000000000001001001100 nv₀ = 0x0000024c nv₀ = x⁹ + x⁶ + x³ + x²
nv₁ = b00000000000000000000000000000001 nv₁ = 0x00000001 nv₁ = 1