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

generates a code of length 29 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+108x^86+96x^87+9x^88+36x^90+6x^100

The gray image is a linear code over GF(4) with n=116, k=4 and d=86.
As d=86 is an upper bound for linear (116,4,4)-codes, this code is optimal over F4[X,sigma]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in -1.01e-007 seconds.