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

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+7x^66+34x^67+175x^68+26x^69+8x^70+2x^71+2x^85+1x^102

The gray image is a linear code over GF(2) with n=544, k=8 and d=264.
This code was found by Heurico 1.16 in 0.219 seconds.