The generator matrix

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

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+81x^16+96x^18+72x^20+6x^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 6.87e-008 seconds.