The generator matrix

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

generates a code of length 75 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+113x^68+630x^69+1427x^70+2218x^71+2673x^72+3772x^73+3624x^74+4354x^75+4081x^76+3202x^77+2361x^78+1958x^79+1048x^80+656x^81+300x^82+186x^83+88x^84+24x^85+18x^86+20x^87+2x^88+4x^89+6x^90+2x^92

The gray image is a linear code over GF(2) with n=600, k=15 and d=272.
This code was found by Heurico 1.16 in 16.3 seconds.