The generator matrix

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

generates a code of length 77 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 220.

Homogenous weight enumerator: w(x)=1x^0+1644x^220+3165x^224+3339x^228+2823x^232+2244x^236+1677x^240+1074x^244+357x^248+48x^252+9x^256+3x^260

The gray image is a linear code over GF(4) with n=308, k=7 and d=220.
This code was found by Heurico 1.16 in 52.1 seconds.