The generator matrix

 1  1  1  1  1  1  1  1  1  X  X X^2  X  1 X^2
 0 X^3+X^2  0  0  0 X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^2
 0  0 X^3+X^2  0 X^2 X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^2  0  0  0 X^3
 0  0  0 X^3+X^2 X^2 X^3 X^3+X^2  0 X^3 X^2 X^3 X^3 X^3 X^3+X^2 X^3+X^2
 0  0  0  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3

generates a code of length 15 over Z2[X]/(X^4) who´s minimum homogenous weight is 11.

Homogenous weight enumerator: w(x)=1x^0+52x^11+72x^12+248x^13+306x^14+680x^15+347x^16+256x^17+12x^18+36x^19+28x^20+8x^21+2x^22

The gray image is a linear code over GF(2) with n=120, k=11 and d=44.
This code was found by Heurico 1.16 in 14.5 seconds.