The generator matrix

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

generates a code of length 52 over F4[X]/(X^2) who´s minimum homogenous weight is 156.

Homogenous weight enumerator: w(x)=1x^0+240x^156+9x^160+6x^176

The gray image is a linear code over GF(4) with n=208, k=4 and d=156.
As d=156 is an upper bound for linear (208,4,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.015 seconds.