The generator matrix

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

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+152x^39+98x^40+310x^41+293x^42+414x^43+295x^44+252x^45+58x^46+116x^47+14x^48+24x^49+1x^50+6x^51+7x^52+4x^53+2x^57+1x^64

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 31.5 seconds.