The generator matrix

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

generates a code of length 69 over Z2[X]/(X^4) who´s minimum homogenous weight is 63.

Homogenous weight enumerator: w(x)=1x^0+490x^63+1336x^64+2194x^65+2778x^66+3556x^67+3916x^68+4802x^69+4056x^70+3296x^71+2428x^72+1808x^73+1062x^74+550x^75+195x^76+154x^77+68x^78+42x^79+26x^80+2x^81+4x^82+2x^83+1x^84+1x^88

The gray image is a linear code over GF(2) with n=552, k=15 and d=252.
This code was found by Heurico 1.16 in 13.9 seconds.