The generator matrix

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

generates a code of length 77 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+336x^72+48x^73+610x^74+184x^75+865x^76+304x^77+732x^78+192x^79+375x^80+32x^81+222x^82+8x^83+138x^84+32x^86+12x^88+4x^90+1x^124

The gray image is a linear code over GF(2) with n=616, k=12 and d=288.
This code was found by Heurico 1.16 in 1.38 seconds.