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

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

Homogenous weight enumerator: w(x)=1x^0+34x^78+8x^79+3x^80+12x^81+2x^82+4x^83

The gray image is a linear code over GF(2) with n=624, k=6 and d=312.
This code was found by Heurico 1.16 in 0.297 seconds.