The generator matrix

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

generates a code of length 11 over Z2[X]/(X^3) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+188x^6+642x^8+384x^9+2228x^10+1280x^11+2264x^12+384x^13+644x^14+165x^16+12x^18

The gray image is a linear code over GF(2) with n=44, k=13 and d=12.
This code was found by Heurico 1.16 in 18.3 seconds.