The generator matrix

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

generates a code of length 28 over F4[X]/(X^2) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+81x^80+576x^82+96x^84+72x^88+192x^90+3x^96+3x^112

The gray image is a linear code over GF(4) with n=112, k=5 and d=80.
This code was found by Heurico 1.16 in 3.24e-008 seconds.