The generator matrix

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

generates a code of length 28 over F4[X]/(X^2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+552x^78+597x^80+1032x^82+372x^84+312x^86+258x^88+792x^90+180x^92

The gray image is a linear code over GF(4) with n=112, k=6 and d=78.
This code was found by Heurico 1.16 in 3.94 seconds.