The generator matrix

 1  0  1  1  1  1  1  1  1  X  1  1  1  1 aX  1  1  1  1 (a+1)X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  X aX (a+1)X  1  1  1  1  1  1  1  1  1  1  1  1
 0  1 (a+1)X+1  a (a+1)X+a+1  X aX+1 X+a aX+a+1  1 aX X+1 aX+a X+a+1  1 (a+1)X  1 (a+1)X+a a+1  1  0 (a+1)X+1  a  X aX+1 X+a (a+1)X+a+1 aX+a+1 aX X+1 aX+a X+a+1 (a+1)X  1 (a+1)X+a a+1  1  1  1  1  0  X aX (a+1)X (a+1)X+1 aX+1 X+1  1  a X+a aX+a (a+1)X+a

generates a code of length 52 over F4[X,sigma]/(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,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.016 seconds.