The generator matrix

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

generates a code of length 79 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+398x^72+1598x^73+2870x^74+4280x^75+5956x^76+6502x^77+7806x^78+7842x^79+7331x^80+6562x^81+5662x^82+3676x^83+2346x^84+1358x^85+746x^86+348x^87+110x^88+72x^89+36x^90+12x^91+18x^92+4x^93+2x^95

The gray image is a linear code over GF(2) with n=632, k=16 and d=288.
This code was found by Heurico 1.16 in 46 seconds.