The generator matrix

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

generates a code of length 10 over Z2[X]/(X^4) who´s minimum homogenous weight is 10.

Homogenous weight enumerator: w(x)=1x^0+26x^10+5x^12

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^4) for dimension 5.
This code was found by Heurico 1.16 in -6.48e-008 seconds.