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

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+612x^78+1080x^79+321x^80+480x^81+2796x^82+2676x^83+1113x^84+1260x^85+5268x^86+5028x^87+2004x^88+2556x^89+8532x^90+7596x^91+2337x^92+2628x^93+8184x^94+5460x^95+1203x^96+756x^97+2256x^98+1200x^99+162x^100+12x^104+9x^108+3x^112+3x^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 11.5 seconds.