The generator matrix

 1  0  0  0  0  1  1  1  0  1 X^2  1  1  1  1  0  0  X  1  1  1 X^2+X X^2+X X^2+X  1 X^2  1  1  1  1  1  1  X X^2+X  1  1  1  1  0 X^2 X^2+X  0  1  1
 0  1  0  0  0  0 X+1  X X^2 X+1  1 X^2 X^2+1 X+1 X^2+X+1  1  1  1 X^2+X X^2+1 X+1  1 X^2  1 X^2+X  0  0 X^2+X  X  0  1  1  1  1 X^2+1  X X^2+X+1  X  1  1  1 X^2  1  0
 0  0  1  0  0  0  1 X+1  1 X^2+1 X^2 X^2+1 X^2+X X^2+X+1 X^2+X  X X+1 X^2+1  1  0 X^2+X X^2+1  X  0 X^2+X  1 X^2  X X^2+1 X^2+X X^2+X  1 X+1 X+1 X^2+X X^2 X^2+X+1 X^2  0 X^2+X+1  1  1 X^2  0
 0  0  0  1  0  1 X^2 X^2+1  1 X+1 X^2+1 X^2+X X^2 X^2+1 X^2+X+1 X^2+X X^2+1  0  0 X^2 X^2+1  X  1  1  X X^2+X X^2+1  1 X^2+X+1  0 X^2+X+1  X X^2  1 X^2+X+1  X X^2 X+1 X^2+X  0 X^2+X X^2 X^2+1  0
 0  0  0  0  1  1 X^2+1  X X+1 X^2+1 X^2+X X^2+1  0 X^2 X^2+X+1 X^2+1 X^2+X+1  0 X^2+X  1  X  1 X^2+1 X+1 X^2+1 X^2+X+1 X^2+1 X^2+X X+1  1 X^2+X X^2+X X^2+X+1  0  1  X  0 X^2  1  0  X  1 X^2  0
 0  0  0  0  0  X  0  X  X X^2+X  X X^2  0  X X^2+X  0 X^2+X X^2 X^2 X^2 X^2+X  0  0 X^2  X X^2+X X^2  0 X^2  X  0 X^2+X  X  0  X  0 X^2+X X^2  X X^2+X X^2 X^2+X X^2+X 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+86x^34+362x^35+900x^36+2082x^37+3101x^38+5300x^39+7470x^40+10340x^41+12618x^42+15612x^43+14832x^44+15528x^45+13048x^46+10988x^47+7671x^48+5176x^49+2773x^50+1698x^51+778x^52+398x^53+179x^54+88x^55+26x^56+12x^57+2x^58+2x^60+1x^66

The gray image is a linear code over GF(2) with n=176, k=17 and d=68.
This code was found by Heurico 1.13 in 119 seconds.