The generator matrix

 1  0  1  1  1
 0  1  1  a a^2*X+a^2
 0  0 a^2*X a*X  X

generates a code of length 5 over F4[X]/(X^2) who´s minimum homogenous weight is 12.

Homogenous weight enumerator: w(x)=1x^0+270x^12+735x^16+18x^20

The gray image is a linear code over GF(4) with n=20, k=5 and d=12.
As d=12 is an upper bound for linear (20,5,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0 seconds.