The generator matrix

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

generates a code of length 16 over F4[X]/(X^2) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+252x^48+3x^64

The gray image is a linear code over GF(4) with n=64, k=4 and d=48.
As d=48 is an upper bound for linear (64,4,4)-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.