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

generates a code of length 43 over Z2[X]/(X^4) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+105x^38+8x^39+140x^40+128x^41+417x^42+496x^43+393x^44+128x^45+125x^46+8x^47+58x^48+23x^50+15x^52+2x^54+1x^72

The gray image is a linear code over GF(2) with n=344, k=11 and d=152.
This code was found by Heurico 1.16 in 1.36 seconds.