The generator matrix

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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 75.

Homogenous weight enumerator: w(x)=1x^0+378x^75+242x^76+344x^77+258x^78+340x^79+155x^80+188x^81+46x^82+74x^83+12x^87+4x^89+4x^91+2x^112

The gray image is a linear code over GF(2) with n=624, k=11 and d=300.
This code was found by Heurico 1.16 in 12.9 seconds.