The generator matrix

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

generates a code of length 29 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+129x^84+576x^85+60x^88+60x^92+192x^93+3x^96+3x^116

The gray image is a linear code over GF(4) with n=116, k=5 and d=84.
As d=84 is an upper bound for linear (116,5,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.031 seconds.