The generator matrix

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

generates a code of length 20 over Z2[X]/(X^3) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+174x^16+672x^20+176x^24+1x^32

The gray image is a linear code over GF(2) with n=80, k=10 and d=32.
This code was found by Heurico 1.16 in 5.7 seconds.