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

generates a code of length 38 over Z2[X]/(X^4) who´s minimum homogenous weight is 32.

Homogenous weight enumerator: w(x)=1x^0+78x^32+163x^36+1632x^38+66x^40+32x^42+52x^44+23x^48+1x^68

The gray image is a linear code over GF(2) with n=304, k=11 and d=128.
This code was found by Heurico 1.16 in 0.469 seconds.