The generator matrix

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

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+80x^39+226x^40+176x^41+480x^42+152x^43+477x^44+144x^45+208x^46+80x^47+12x^48+8x^51+3x^52+1x^56

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 0.078 seconds.