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+1)X+a+1  0 (a+1)X+1  a  X (a+1)X+a+1  1 (a+1)X+1 X+a aX+a+1  1  X (a+1)X (a+1)X aX+1 aX+1  1 X+a (a+1)X+a (a+1)X+a aX+a+1  1 a+1 a+1
 0  0 (a+1)X  X aX  X  0 (a+1)X aX (a+1)X (a+1)X  X aX  0 aX  X aX  0 (a+1)X  0  X (a+1)X aX  X (a+1)X (a+1)X aX  0

generates a code of length 28 over F4[X,sigma]/(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 0.016 seconds.