The generator matrix

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

generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 63.

Homogenous weight enumerator: w(x)=1x^0+80x^63+635x^64+954x^65+1171x^66+1112x^67+1097x^68+946x^69+718x^70+446x^71+337x^72+248x^73+268x^74+106x^75+36x^76+12x^77+18x^78+5x^80+1x^82+1x^84

The gray image is a linear code over GF(2) with n=544, k=13 and d=252.
This code was found by Heurico 1.16 in 9.75 seconds.