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

generates a code of length 44 over Z2[X]/(X^3) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+127x^34+486x^36+8x^37+866x^38+184x^39+1330x^40+680x^41+2051x^42+1176x^43+2479x^44+1176x^45+2173x^46+680x^47+1421x^48+184x^49+775x^50+8x^51+392x^52+144x^54+32x^56+6x^58+3x^60+1x^62+1x^74

The gray image is a linear code over GF(2) with n=176, k=14 and d=68.
This code was found by Heurico 1.16 in 17.3 seconds.