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  X  1  X  X  X  1  1  1  X  X  1  X  X  X  1  X  X  X  X  X  1  1  1 X^2  0 X^2  0  X  X X^2  0 X^2 X^3 X^2  0 X^2 X^3 X^2 X^3 X^2 X^3  1  1  1  1
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^2 X^3  0  0 X^3  0 X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^2 X^2 X^3 X^2 X^3+X^2 X^2 X^3+X^2 X^2  0 X^3 X^3+X^2 X^2 X^2 X^2 X^3+X^2 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0  0  0 X^3
 0  0 X^3  0  0 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  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0  0  0 X^3 X^3
 0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0  0  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 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+8x^66+234x^68+8x^70+1x^72+1x^80+2x^84+1x^88

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 0.281 seconds.