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  1  1  1  1  1  1  1  1  1  1  1  1  0  0  X X^2  1  0  1
 0  X  0  0  0  0  0  0  0  0 X^2  0 X^2 X^2  0  0 X^2 X^2 X^2  0 X^2 X^2  X  X  X X^2+X X^2+X  X  X  X X^2+X X^2+X  X  X  0  X  X  X X^2 X^2+X X^2 X^2  X X^2
 0  0  X  0  0  0  0  0  0  0 X^2+X  X  X  X  X X^2+X  X  0 X^2 X^2  0 X^2+X X^2+X X^2+X  X X^2  0 X^2 X^2  0 X^2+X  X  X  X X^2  0  0 X^2  0 X^2  X  X  0  0
 0  0  0  X  0  0  0  X X^2+X  X  X  X  0  X X^2  X  0  0 X^2+X X^2 X^2 X^2 X^2+X  X  X  X X^2  0 X^2+X X^2  0 X^2+X  X  0  0  X  0  X  X X^2  0  0 X^2+X  0
 0  0  0  0  X  0  X  X  X X^2  X X^2  X X^2+X X^2 X^2 X^2+X  X X^2+X  X X^2 X^2 X^2  0 X^2+X X^2+X  0  X X^2+X X^2+X  0  X  X  0  X  0  X X^2  0  0 X^2+X  X  0  0
 0  0  0  0  0  X  X X^2 X^2+X X^2+X  X  0  X  0  X X^2+X  0 X^2+X X^2+X  0  0 X^2 X^2  X  0  0  0 X^2 X^2  X X^2+X X^2  X  X X^2+X X^2+X X^2+X  X  0  0 X^2 X^2  X X^2+X
 0  0  0  0  0  0 X^2 X^2 X^2 X^2 X^2  0 X^2  0 X^2  0 X^2  0  0 X^2 X^2 X^2  0  0 X^2  0 X^2  0 X^2  0  0 X^2 X^2 X^2  0  0 X^2 X^2  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+78x^34+130x^35+219x^36+316x^37+375x^38+466x^39+555x^40+860x^41+1496x^42+2306x^43+2748x^44+2328x^45+1483x^46+928x^47+609x^48+464x^49+328x^50+234x^51+201x^52+124x^53+77x^54+30x^55+19x^56+4x^57+2x^58+2x^59+1x^78

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 11.4 seconds.