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

generates a code of length 67 over Z2[X]/(X^3) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+40x^56+62x^57+129x^58+172x^59+259x^60+314x^61+394x^62+484x^63+552x^64+616x^65+749x^66+826x^67+717x^68+690x^69+498x^70+378x^71+296x^72+300x^73+195x^74+146x^75+155x^76+60x^77+65x^78+42x^79+23x^80+6x^81+13x^82+5x^84+2x^86+2x^90+1x^94

The gray image is a linear code over GF(2) with n=268, k=13 and d=112.
This code was found by Heurico 1.16 in 5.98 seconds.