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  0 X^2  X  X  X X^2  0 X^2  1  0  X  X  1  0  1  X  0  1  1
 0  X  0  0  0  0  0  0  0  X X^2+X  X  X X^2+X  X X^2  X X^2 X^2 X^2+X  0 X^2 X^2+X  X X^2 X^2  X  X X^2 X^2  X  X  X  X  X X^2+X  X X^2  0  X X^2+X  X  X  0
 0  0  X  0  0  0  X X^2+X  X  0  0  0  X  X X^2+X X^2  X  X X^2+X X^2 X^2+X  0 X^2 X^2+X X^2  X  X X^2  0  X X^2+X  0  0  0  X  X X^2+X  X  X X^2+X X^2+X X^2 X^2+X  0
 0  0  0  X  0  X  X X^2+X  0  X  X X^2  0 X^2 X^2+X  X X^2+X  0 X^2+X X^2+X  X X^2+X  X X^2 X^2+X  0  X X^2  X  X  0  X X^2+X  0  0  0 X^2+X  X X^2+X  0  0 X^2+X X^2  X
 0  0  0  0  X  X  0 X^2+X  X X^2 X^2+X X^2+X  0 X^2+X  X X^2  0  X  X X^2  0 X^2+X X^2+X X^2+X  0  0 X^2+X  X X^2  X X^2  X X^2+X X^2 X^2+X X^2+X X^2  X  X X^2+X X^2 X^2  X  X
 0  0  0  0  0 X^2  0  0  0  0  0  0 X^2  0  0 X^2 X^2 X^2 X^2 X^2 X^2  0  0 X^2  0 X^2  0 X^2 X^2  0 X^2  0 X^2  0 X^2 X^2  0 X^2  0  0 X^2 X^2  0 X^2
 0  0  0  0  0  0 X^2  0 X^2 X^2  0 X^2  0  0 X^2 X^2 X^2  0 X^2  0  0 X^2  0 X^2  0 X^2 X^2  0 X^2  0  0  0 X^2  0 X^2 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2
 0  0  0  0  0  0  0 X^2 X^2  0 X^2 X^2 X^2 X^2  0 X^2  0 X^2  0 X^2 X^2  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2 X^2  0 X^2  0  0 X^2 X^2 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+69x^34+136x^35+235x^36+330x^37+525x^38+700x^39+922x^40+1146x^41+1438x^42+1814x^43+1781x^44+1704x^45+1516x^46+1238x^47+931x^48+678x^49+457x^50+314x^51+207x^52+100x^53+87x^54+22x^55+18x^56+8x^57+3x^58+1x^60+2x^61+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 11.7 seconds.