gf(2) Find null basis vectors of M

M = 000000000000000000
    000000000000000110
    000000000000010100
    000000000001001000
    000000000100010000
    000000010000100000
    000001000001000000
    000100000010000000
    010000000100000000
    101010100111101101
    010101011110000011
    111110110111100001
    101101000001011110
    001101110101001111
    111011010100111100
    100000001100101010
    001111000010011111
    110110100110010001
Augment M with identity (I) of same size
M = 000000000000000000 000000000000000001
    000000000000000110 000000000000000010
    000000000000010100 000000000000000100
    000000000001001000 000000000000001000
    000000000100010000 000000000000010000
    000000010000100000 000000000000100000
    000001000001000000 000000000001000000
    000100000010000000 000000000010000000
    010000000100000000 000000000100000000
    101010100111101101 000000001000000000
    010101011110000011 000000010000000000
    111110110111100001 000000100000000000
    101101000001011110 000001000000000000
    001101110101001111 000010000000000000
    111011010100111100 000100000000000000
    100000001100101010 001000000000000000
    001111000010011111 010000000000000000
    110110100110010001 100000000000000000
M expressed in convenient bit depth
M = 0000000000000000000000000000000000000000000000000000000000000001
    0000000000000000000000000000000000000000001100000000000000000010
    0000000000000000000000000000000000000000101000000000000000000100
    0000000000000000000000000000000000000010010000000000000000001000
    0000000000000000000000000000000000001000100000000000000000010000
    0000000000000000000000000000000000100001000000000000000000100000
    0000000000000000000000000000000010000010000000000000000001000000
    0000000000000000000000000000001000000100000000000000000010000000
    0000000000000000000000000000100000001000000000000000000100000000
    0000000000000000000000000001010101001111011010000000001000000000
    0000000000000000000000000000101010111100000110000000010000000000
    0000000000000000000000000001111101101111000010000000100000000000
    0000000000000000000000000001011010000010111100000001000000000000
    0000000000000000000000000000011011101010011110000010000000000000
    0000000000000000000000000001110110101001111000000100000000000000
    0000000000000000000000000001000000011001010100001000000000000000
    0000000000000000000000000000011110000100111110010000000000000000
    0000000000000000000000000001101101001100100010100000000000000000
Transform M into row reduced echelon form (rref)
M = 0000000000000000000000000001000000000000000110001000010111111110
    0000000000000000000000000000100000000000000100000000000100010110
    0000000000000000000000000000010000000000010100000110001110000100
    0000000000000000000000000000001000000000000110001111011000111110
    0000000000000000000000000000000100000000010010001010011100010110
    0000000000000000000000000000000010000000010000000000000001001000
    0000000000000000000000000000000001000000010010000100100011000100
    0000000000000000000000000000000000100000010110001111110100101010
    0000000000000000000000000000000000010000000000001111100011100010
    0000000000000000000000000000000000001000000100000000000000010110
    0000000000000000000000000000000000000100000110001111011010111110
    0000000000000000000000000000000000000010010000000000000000001000
    0000000000000000000000000000000000000001010110001111110100001010
    0000000000000000000000000000000000000000100100000000000000000110
    0000000000000000000000000000000000000000001100000000000000000010
    0000000000000000000000000000000000000000000000100110101110101010
    0000000000000000000000000000000000000000000000011100010001011110
    0000000000000000000000000000000000000000000000000000000000000001
M in augmented form
M = 100000000000000011 001000010111111110
    010000000000000010 000000000100010110
    001000000000001010 000110001110000100
    000100000000000011 001111011000111110
    000010000000001001 001010011100010110
    000001000000001000 000000000001001000
    000000100000001001 000100100011000100
    000000010000001011 001111110100101010
    000000001000000000 001111100011100010
    000000000100000010 000000000000010110
    000000000010000011 001111011010111110
    000000000001001000 000000000000001000
    000000000000101011 001111110100001010
    000000000000010010 000000000000000110
    000000000000000110 000000000000000010
    000000000000000000 100110101110101010
    000000000000000000 011100010001011110
    000000000000000000 000000000000000001
Highlight null basis vectors (where left side of row is zero)
M = 100000000000000011 001000010111111110
    010000000000000010 000000000100010110
    001000000000001010 000110001110000100
    000100000000000011 001111011000111110
    000010000000001001 001010011100010110
    000001000000001000 000000000001001000
    000000100000001001 000100100011000100
    000000010000001011 001111110100101010
    000000001000000000 001111100011100010
    000000000100000010 000000000000010110
    000000000010000011 001111011010111110
    000000000001001000 000000000000001000
    000000000000101011 001111110100001010
    000000000000010010 000000000000000110
    000000000000000110 000000000000000010
    000000000000000000 100110101110101010
    000000000000000000 011100010001011110
    000000000000000000 000000000000000001
Null basis vectors

nv0 = b0000000000000000000000000000000000000000000000100110101110101010
nv0 = 0x0000000000026baa
nv0 = x17 + x14 + x13 + x11 + x9 + x8 + x7 + x5 + x3 + x

nv1 = b0000000000000000000000000000000000000000000000011100010001011110
nv1 = 0x000000000001c45e
nv1 = x16 + x15 + x14 + x10 + x6 + x4 + x3 + x2 + x

nv2 = b0000000000000000000000000000000000000000000000000000000000000001
nv2 = 0x0000000000000001
nv2 = 1

See also: