The generator matrix

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

generates a code of length 17 over F4[X]/(X^2) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+432x^48+432x^52+144x^56+15x^64

The gray image is a linear code over GF(4) with n=68, k=5 and d=48.
As d=48 is an upper bound for linear (68,5,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.016 seconds.