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

generates a code of length 30 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+135x^84+366x^88+468x^92+21x^96+6x^100+6x^104+12x^108+6x^112+3x^116

The gray image is a linear code over GF(4) with n=120, k=5 and d=84.
This code was found by Heurico 1.16 in 0.75 seconds.