The generator matrix

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

generates a code of length 30 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+804x^78+1104x^79+171x^80+3396x^82+2748x^83+660x^84+7104x^86+5520x^87+1185x^88+12192x^90+8328x^91+1236x^92+10428x^94+5568x^95+744x^96+2940x^98+1308x^99+66x^100+27x^104+6x^116

The gray image is a linear code over GF(4) with n=120, k=8 and d=78.
This code was found by Heurico 1.16 in 14.7 seconds.