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

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+52x^38+90x^40+32x^41+152x^42+352x^43+727x^44+352x^45+146x^46+32x^47+49x^48+22x^50+24x^52+10x^54+4x^56+2x^58+1x^76

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 0.141 seconds.