The generator matrix

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

generates a code of length 29 over F4[X]/(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]/(X^2) for dimension 4.
This code was found by Heurico 1.16 in 0.016 seconds.