The generator matrix

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

generates a code of length 49 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 144.

Homogenous weight enumerator: w(x)=1x^0+489x^144+396x^148+18x^152+108x^156+12x^160

The gray image is a linear code over GF(4) with n=196, k=5 and d=144.
As d=144 is an upper bound for linear (196,5,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.375 seconds.