The generator matrix

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

generates a code of length 39 over Z2[X]/(X^4) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+27x^40+4x^42

The gray image is a linear code over GF(2) with n=312, k=5 and d=160.
This code was found by Heurico 1.16 in 0.015 seconds.