The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X X^3  1  1  1  1  X  1  1  1  0  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X X^2 X^2+X  0 X^2+X X^3 X^3+X X^2  X X^2+X  X  0 X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^2  0  X X^3+X X^3+X^2+X  0 X^2+X  X  X X^3  0 X^3+X^2 X^2 X^3+X^2+X X^3+X^2+X X^3+X  X  X X^3
 0  0 X^3+X^2  0 X^2 X^2 X^3 X^2 X^2  0 X^3 X^3+X^2 X^2 X^2 X^3 X^3  0 X^3+X^2  0  0 X^3 X^3 X^3+X^2 X^2 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3+X^2 X^2 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2 X^2 X^3+X^2 X^3  0  0  0 X^3 X^3
 0  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0  0  0  0  0 X^3 X^3 X^3
 0  0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0  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+130x^39+84x^40+206x^41+434x^42+390x^43+451x^44+174x^45+12x^46+86x^47+39x^48+34x^49+2x^50+2x^51+2x^53+1x^76

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 134 seconds.