The generator matrix

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

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+281x^68+1328x^69+3311x^70+5382x^71+7828x^72+10578x^73+13416x^74+15076x^75+16577x^76+15272x^77+13785x^78+10982x^79+7847x^80+4482x^81+2538x^82+1400x^83+590x^84+196x^85+94x^86+56x^87+24x^88+16x^89+4x^90+2x^92+4x^94+2x^96

The gray image is a linear code over GF(2) with n=608, k=17 and d=272.
This code was found by Heurico 1.16 in 176 seconds.