The generator matrix

 1  0  0  1  1  1  0 X^3 X^2 X^3+X^2  1  1  1  1 X^3+X  1 X^3+X  1  1  1 X^2+X X^3+X^2+X  1 X^3+X  1  0  1  1 X^2  1 X^3+X X^3+X^2+X  0  1 X^2  1  1  1  1  1  1  1  1
 0  1  0  0 X^2+1 X^2+1  1 X^3+X^2+X  1  1 X^3+X^2  1 X^3+X X^2+X+1 X^3+X^2+X X^2+X  1 X^3+X^2+X+1 X^3+X^2+X X^3+1  1 X^2 X^2  1 X^3+X^2+1  1 X+1 X^3+X^2+X X^2+X X+1  0  1  1 X^3 X^2 X^2+X X^3  0 X^3+X  0 X^2+X+1 X^3+X^2+1  X
 0  0  1 X+1 X^3+X+1 X^3 X^3+X^2+X+1  1  X X^3+1 X^2+X  1 X^2+1  X  1 X^3+X^2+X+1 X+1 X+1 X^3 X^3+X^2+X X^3+X^2  1 X^3+1 X^3+X^2+1  1 X^2 X^3+X^2+1 X^3+X^2  1 X^3+X^2+X+1  1 X^3+1  X X^3  1 X+1  1 X^2+1 X^2+X X^3+X^2+X+1 X^3+X X^3+X  X
 0  0  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3 X^3  0  0  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 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+340x^39+660x^40+1444x^41+1144x^42+1476x^43+1042x^44+940x^45+480x^46+444x^47+115x^48+76x^49+8x^50+12x^51+6x^52+4x^53

The gray image is a linear code over GF(2) with n=344, k=13 and d=156.
This code was found by Heurico 1.16 in 0.891 seconds.