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

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+100x^76+64x^77+198x^78+64x^79+67x^80+8x^82+8x^84+2x^110

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 0.968 seconds.