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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+63x^38+107x^40+56x^42+256x^43+1088x^44+256x^45+107x^46+70x^48+8x^50+21x^54+12x^56+1x^62+1x^64+1x^72

The gray image is a linear code over GF(2) with n=352, k=11 and d=152.
This code was found by Heurico 1.16 in 97.6 seconds.