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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+113x^76+112x^77+108x^78+64x^79+72x^80+16x^81+20x^82+2x^84+1x^88+1x^92+1x^96+1x^104

The gray image is a linear code over GF(2) with n=624, k=9 and d=304.
This code was found by Heurico 1.16 in 0.5 seconds.