The generator matrix

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

generates a code of length 16 over F4[X]/(X^2) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+1440x^44+1212x^48+1440x^52+3x^64

The gray image is a linear code over GF(4) with n=64, k=6 and d=44.
As d=44 is an upper bound for linear (64,6,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 6.
This code was found by Heurico 1.16 in 8.67 seconds.