The generator matrix

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

generates a code of length 34 over Z2[X]/(X^3) who´s minimum homogenous weight is 26.

Homogenous weight enumerator: w(x)=1x^0+139x^26+610x^27+1194x^28+2362x^29+3509x^30+5248x^31+6470x^32+8458x^33+8855x^34+8866x^35+7024x^36+5668x^37+3297x^38+1948x^39+1036x^40+522x^41+190x^42+96x^43+18x^44+14x^45+10x^46+1x^48

The gray image is a linear code over GF(2) with n=136, k=16 and d=52.
This code was found by Heurico 1.13 in 21.7 seconds.