The generator matrix

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

generates a code of length 28 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+528x^78+597x^80+1104x^82+372x^84+240x^86+258x^88+816x^90+180x^92

The gray image is a linear code over GF(4) with n=112, k=6 and d=78.
This code was found by Heurico 1.16 in 0.062 seconds.