The generator matrix

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

generates a code of length 43 over Z2[X]/(X^4) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+56x^39+216x^40+244x^41+413x^42+380x^43+337x^44+124x^45+102x^46+52x^47+69x^48+32x^49+13x^50+8x^51+1x^68

The gray image is a linear code over GF(2) with n=344, k=11 and d=156.
This code was found by Heurico 1.16 in 0.094 seconds.