The generator matrix

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

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+113x^68+4x^70+4x^72+4x^74+1x^76+1x^80

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 0.171 seconds.