The generator matrix

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

generates a code of length 27 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+120x^78+45x^80+72x^86+18x^88

The gray image is a linear code over GF(4) with n=108, k=4 and d=78.
This code was found by Heurico 1.16 in -3.62e-008 seconds.