The generator matrix

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

generates a code of length 6 over F4[X]/(X^2) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+45x^16+192x^18+18x^24

The gray image is a linear code over GF(4) with n=24, k=4 and d=16.
As d=16 is an upper bound for linear (24,4,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in -1.01e-007 seconds.