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

generates a code of length 20 over Z2[X]/(X^4) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+174x^16+1696x^20+176x^24+1x^32

The gray image is a linear code over GF(2) with n=160, k=11 and d=64.
This code was found by Heurico 1.16 in 18.6 seconds.