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

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+52x^66+160x^67+597x^68+160x^69+52x^70+1x^72+1x^132

The gray image is a linear code over GF(2) with n=544, k=10 and d=264.
This code was found by Heurico 1.16 in 0.422 seconds.