The generator matrix

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

generates a code of length 78 over Z2[X]/(X^3) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+234x^72+278x^73+471x^74+304x^75+496x^76+310x^77+421x^78+190x^79+397x^80+200x^81+217x^82+100x^83+114x^84+78x^85+103x^86+38x^87+64x^88+30x^89+32x^90+8x^91+2x^92+4x^94+4x^96

The gray image is a linear code over GF(2) with n=312, k=12 and d=144.
This code was found by Heurico 1.16 in 0.899 seconds.