The generator matrix

 1  0  0  0  1  1  1 X^2  1 X^3  1  1 X^2  1 X^2  1  0  1 X^2+X  1  1  1 X^2+X  1 X^3+X X^3  1  1  1  1  X  1  1 X^3+X^2+X  1 X^3+X X^3+X^2+X  0  0  1  1 X^3+X^2  1 X^3+X^2+X  0 X^2+X X^3+X^2 X^3+X X^3+X  1  1  1  1  1  1  1 X^2+X  1 X^3+X X^3+X^2+X  1  1  X  1 X^3 X^2+X X^3+X^2  1
 0  1  0  0 X^3  1 X^3+1  1 X^2  1 X^3+X X^3+X+1 X^3+X X^2+X+1  1  X X^3+X X^3+X^2+1  1 X^3+1 X^3+X X^3+X^2+X  1 X^3+X^2+X X^2 X^3+X X^2+1 X^2 X^2+1 X^3+X^2  1 X^3 X^3+1  1 X^2+X X^2  1 X^3  1 X^3+X+1 X+1  1 X^3+X^2+X X^2+X  1  1 X^3+X^2  1 X^2+X X^3 X^2 X^3+1 X^2+X+1 X^2+1 X^3 X^2+X+1  1  X X^2 X^2+X X^2+X X^2 X^3+X^2+X  X X^2+X  1  1 X^3
 0  0  1  0 X^3+1  1 X^3 X^3+X^2+1  0 X^3+X^2 X^2+1 X^3+X^2  1 X^3+X^2+X+1 X^3+X+1 X+1  1 X^2+X+1 X+1 X^3+X+1 X^3+1 X^2 X^3+X X^3+X^2+X  1 X^2+X X^2+X X^2+X  0  1 X^3+X^2+1  1 X^3+X^2+X+1 X^3+X  0  1 X^2+1  1 X^3+X^2+X X^2+X X^3 X^2 X^3+1 X^2 X^3+X^2+1 X^3+X^2  1 X^3+X^2+X+1  1  0  X  X X^3+1 X^3+X^2+X X^2 X^2+X+1 X^3+1 X^3+X^2+X+1 X^3 X^3+X X^3+X+1 X^3+X  1 X^2+1 X^2 X^2+X X^3+X X^2
 0  0  0  1  1 X^3 X^3+X^2+1 X^3+X^2+1 X^3+1 X^3+1  0 X^2 X^2+X+1 X^2+1 X^2 X^3+X^2+X+1 X^2+X X^3+X X^3+X^2+1 X^3+X+1  X X+1 X^3+1 X^2+X X^2+1  1 X^3 X^2+X+1 X^3+X^2+X+1 X^3+X^2+X  0 X^2+X+1  0 X^2  X X^2+X+1 X^3+X^2+X+1 X^3+X^2+X X^3+X+1 X^3+X+1  X X^2+X+1 X^2+X+1  1  X X^3+X^2 X^3+X^2+1 X^2+X+1 X^2 X^3+X+1  1 X^2+X+1  0 X^3+X^2+1 X^3+X^2+X X^3+X+1 X^2+X X^3+X  1  1 X^3+X^2+1 X^3+X^2 X^3+X^2+1  X  1 X^3+X^2 X^3+X^2  0

generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 61.

Homogenous weight enumerator: w(x)=1x^0+126x^61+1296x^62+2566x^63+4133x^64+5460x^65+6928x^66+7936x^67+8791x^68+8190x^69+7108x^70+5338x^71+3816x^72+2032x^73+1052x^74+360x^75+211x^76+96x^77+56x^78+20x^79+16x^80+4x^83

The gray image is a linear code over GF(2) with n=544, k=16 and d=244.
This code was found by Heurico 1.16 in 36.8 seconds.