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

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

Homogenous weight enumerator: w(x)=1x^0+501x^144+360x^148+54x^152+96x^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]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.359 seconds.