The generator matrix

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

generates a code of length 98 over F4[X]/(X^2) who´s minimum homogenous weight is 290.

Homogenous weight enumerator: w(x)=1x^0+228x^290+168x^291+18x^292+144x^294+144x^295+30x^296+180x^298+72x^299+24x^302+3x^304+3x^308+6x^312+3x^324

The gray image is a linear code over GF(4) with n=392, k=5 and d=290.
This code was found by Heurico 1.16 in 0.188 seconds.