The generator matrix

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

generates a code of length 78 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 222.

Homogenous weight enumerator: w(x)=1x^0+672x^222+552x^223+120x^224+1560x^226+1080x^227+321x^228+1992x^230+1512x^231+117x^232+1524x^234+1056x^235+126x^236+1332x^238+792x^239+174x^240+1032x^242+588x^243+69x^244+744x^246+444x^247+57x^248+300x^250+108x^251+27x^252+60x^254+12x^255+3x^256+6x^260+3x^268

The gray image is a linear code over GF(4) with n=312, k=7 and d=222.
This code was found by Heurico 1.16 in 1.48 seconds.