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

generates a code of length 79 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+30x^78+448x^79+30x^80+1x^94+1x^96+1x^126

The gray image is a linear code over GF(2) with n=632, k=9 and d=312.
This code was found by Heurico 1.16 in 2.34 seconds.