The generator matrix

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

generates a code of length 68 over Z2[X]/(X^4) who´s minimum homogenous weight is 62.

Homogenous weight enumerator: w(x)=1x^0+598x^62+952x^63+2533x^64+2344x^65+4356x^66+3648x^67+4673x^68+3640x^69+3656x^70+2296x^71+2173x^72+856x^73+670x^74+80x^75+205x^76+8x^77+62x^78+13x^80+2x^82+2x^84

The gray image is a linear code over GF(2) with n=544, k=15 and d=248.
This code was found by Heurico 1.16 in 26.1 seconds.