The generator matrix

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

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+376x^11+1888x^12+5790x^13+15130x^14+18954x^15+15555x^16+5692x^17+1700x^18+412x^19+28x^20+6x^21+2x^22+2x^23

The gray image is a linear code over GF(2) with n=120, k=16 and d=44.
This code was found by Heurico 1.16 in 6.59 seconds.