The generator matrix

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

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+60x^37+47x^38+16x^39+101x^40+140x^41+366x^42+608x^43+376x^44+124x^45+92x^46+16x^47+29x^48+52x^49+2x^50+8x^53+5x^54+4x^56+1x^72

The gray image is a linear code over GF(2) with n=344, k=11 and d=148.
This code was found by Heurico 1.16 in 3.3 seconds.