The generator matrix

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

generates a code of length 16 over Z2[X]/(X^4) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+82x^12+16x^13+176x^14+368x^15+770x^16+368x^17+168x^18+16x^19+73x^20+8x^22+1x^24+1x^28

The gray image is a linear code over GF(2) with n=128, k=11 and d=48.
This code was found by Heurico 1.16 in 0.016 seconds.