The generator matrix

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

generates a code of length 30 over F4[X]/(X^2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+732x^78+1008x^79+225x^80+3444x^82+3084x^83+612x^84+6984x^86+5616x^87+1212x^88+12120x^90+7560x^91+1224x^92+11100x^94+5952x^95+603x^96+2484x^98+1356x^99+180x^100+36x^104+3x^112

The gray image is a linear code over GF(4) with n=120, k=8 and d=78.
This code was found by Heurico 1.16 in 37.6 seconds.