The generator matrix

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

generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 63.

Homogenous weight enumerator: w(x)=1x^0+212x^63+238x^64+426x^65+386x^66+570x^67+646x^68+528x^69+312x^70+356x^71+121x^72+126x^73+46x^74+54x^75+28x^76+24x^77+8x^78+8x^79+4x^80+1x^84+1x^108

The gray image is a linear code over GF(2) with n=544, k=12 and d=252.
This code was found by Heurico 1.16 in 102 seconds.