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

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

Homogenous weight enumerator: w(x)=1x^0+120x^68+1x^72+4x^76+1x^80+1x^88

The gray image is a linear code over GF(2) with n=544, k=7 and d=272.
This code was found by Heurico 1.16 in 24.5 seconds.