The generator matrix

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

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+189x^80+768x^84+63x^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 0.343 seconds.