The generator matrix

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

generates a code of length 30 over F4[X]/(X^2) who´s minimum homogenous weight is 88.

Homogenous weight enumerator: w(x)=1x^0+765x^88+255x^96+3x^120

The gray image is a linear code over GF(4) with n=120, k=5 and d=88.
As d=88 is an upper bound for linear (120,5,4)-codes, this code is optimal over F4[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.031 seconds.