The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X  1  1  1  1
 0 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 X^3
 0  0 X^3  0  0  0  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0  0
 0  0  0 X^3  0  0  0 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0
 0  0  0  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0
 0  0  0  0  0 X^3  0 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0
 0  0  0  0  0  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0 X^3 X^3 X^3  0  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+57x^16+32x^18+853x^20+32x^22+38x^24+10x^28+1x^36

The gray image is a linear code over GF(2) with n=160, k=10 and d=64.
This code was found by Heurico 1.16 in 0.016 seconds.