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

generates a code of length 68 over Z2[X]/(X^3) who´s minimum homogenous weight is 60.

Homogenous weight enumerator: w(x)=1x^0+263x^60+24x^61+114x^62+140x^63+416x^64+284x^65+130x^66+548x^67+337x^68+612x^69+114x^70+308x^71+283x^72+100x^73+90x^74+28x^75+175x^76+4x^77+52x^78+51x^80+12x^82+9x^84+1x^104

The gray image is a linear code over GF(2) with n=272, k=12 and d=120.
This code was found by Heurico 1.16 in 12.3 seconds.