The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+28x^20+2x^24+1x^32

The gray image is a linear code over GF(2) with n=80, k=5 and d=40.
As d=40 is an upper bound for linear (80,5,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 5.
This code was found by Heurico 1.16 in 0.00158 seconds.