The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+57x^20+4x^22+1x^24+1x^28

The gray image is a linear code over GF(2) with n=160, k=6 and d=80.
As d=80 is an upper bound for linear (160,6,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 6.
This code was found by Heurico 1.16 in 6.87e-008 seconds.