The generator matrix

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

generates a code of length 34 over Z2[X]/(X^3) who´s minimum homogenous weight is 27.

Homogenous weight enumerator: w(x)=1x^0+244x^27+678x^28+1206x^29+1872x^30+2688x^31+3407x^32+3904x^33+4362x^34+4300x^35+3668x^36+2744x^37+1790x^38+1000x^39+488x^40+264x^41+102x^42+24x^43+14x^44+10x^45+2x^46

The gray image is a linear code over GF(2) with n=136, k=15 and d=54.
This code was found by Heurico 1.16 in 17.3 seconds.