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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+30x^77+207x^78+15x^80+2x^93+1x^126

The gray image is a linear code over GF(2) with n=624, k=8 and d=308.
This code was found by Heurico 1.16 in 13.5 seconds.