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

generates a code of length 30 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+348x^80+192x^84+1392x^88+1728x^92+285x^96+120x^104+30x^112

The gray image is a linear code over GF(4) with n=120, k=6 and d=80.
This code was found by Heurico 1.16 in 89.7 seconds.