The generator matrix

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

generates a code of length 12 over F4[X]/(X^2) who´s minimum homogenous weight is 35.

Homogenous weight enumerator: w(x)=1x^0+144x^35+60x^36+48x^39+3x^48

The gray image is a linear code over GF(4) with n=48, k=4 and d=35.
As d=35 is an upper bound for linear (48,4,4)-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.