The generator matrix

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

generates a code of length 43 over Z2[X]/(X^4) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+204x^37+1406x^38+2844x^39+4641x^40+8146x^41+9896x^42+11478x^43+9654x^44+8356x^45+4869x^46+2274x^47+1168x^48+436x^49+98x^50+42x^51+8x^52+8x^53+3x^54+2x^55+2x^57

The gray image is a linear code over GF(2) with n=344, k=16 and d=148.
This code was found by Heurico 1.16 in 26.5 seconds.