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

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

Homogenous weight enumerator: w(x)=1x^0+24x^78+7x^80

The gray image is a linear code over GF(2) with n=608, k=5 and d=312.
This code was found by Heurico 1.16 in 0.265 seconds.