The generator matrix

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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+136x^72+804x^73+1127x^74+1902x^75+1528x^76+2232x^77+1864x^78+1898x^79+1398x^80+1376x^81+649x^82+668x^83+384x^84+248x^85+50x^86+70x^87+21x^88+12x^89+2x^90+4x^91+2x^92+4x^94+2x^96+2x^99

The gray image is a linear code over GF(2) with n=624, k=14 and d=288.
This code was found by Heurico 1.16 in 7.72 seconds.