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

generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+212x^66+136x^68+126x^70+18x^72+8x^74+2x^76+6x^78+1x^80+2x^84

The gray image is a linear code over GF(2) with n=544, k=9 and d=264.
This code was found by Heurico 1.16 in 58.7 seconds.