The generator matrix

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

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+44x^12+16x^13+42x^14+48x^15+719x^16+48x^17+52x^18+16x^19+32x^20+2x^22+4x^24

The gray image is a linear code over GF(2) with n=128, k=10 and d=48.
This code was found by Heurico 1.16 in 0.015 seconds.