The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 0  X  0  0  0  0  0  0  0  0  0  0  0  X  0  0
 0  0  X  0  0  0  0  0  0  0  0  0  0  X  0  0
 0  0  0  X  0  0  0  0  0  0  0  0  0  X  0  0
 0  0  0  0  X  0  0  0  0  0  0  0  0  X  0  0
 0  0  0  0  0  X  0  0  0  0  0  0  0  X  0  0
 0  0  0  0  0  0  X  0  0  0  0  0  0  X  0  0
 0  0  0  0  0  0  0  X  0  0  0  0  0  X  0  0
 0  0  0  0  0  0  0  0  X  0  0  0  0  X  0  0
 0  0  0  0  0  0  0  0  0  X  0  0  0  X  0  0
 0  0  0  0  0  0  0  0  0  0  X  0  0  X  0  0
 0  0  0  0  0  0  0  0  0  0  0  X  0  X  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  X  X  0  0

generates a code of length 16 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+78x^4+728x^8+2002x^12+10766x^16+2002x^20+728x^24+78x^28+1x^32

The gray image is a linear code over GF(2) with n=32, k=14 and d=4.
This code was found by Heurico 1.16 in 0.75 seconds.