The generator matrix

 1  0  1  1  1  1  1  1  1  0  1  1  1  0  1  1  1  1  1  X a^2*X  1  1  1  1  1  1  1  1  1  1  1  0  1  1 a*X  1  1  1  1  1  1  1  1  1  X  X  1  1  1  1  1  1  1 a^2*X  1  0  1  1  1  1  0 a*X  1  X  1  1  1  X  X  1  1  X  1  X  1  1  1
 0  1  1  a a^2  0 a^2*X+1 a^2*X+a^2  a  1  0 a^2*X+1  a  1 a^2*X+a^2  X X+a a^2*X+a^2 a^2*X+1  1  1  a a^2*X+a^2  1 a^2*X+a a^2  0  a a*X+1  0 a*X+a^2 a^2*X  1 a^2*X+a^2 a^2*X+a  1 a^2 a^2*X+1 a^2*X+a X+a^2 a*X a^2*X X+a X+1 a^2  1  1 a*X+1  1  a a^2*X  X a*X a*X+a  1 a^2  1 a*X+1 a^2*X a*X+1 a*X+a  1  1 X+a^2  1 a*X+a a*X a*X+1  1  1 a^2*X a^2*X  1  X  1 a*X a*X+1 a^2*X+a^2
 0  0 a^2*X  0  0  0  X  X  X  X  X  X a^2*X a^2*X a*X  X a^2*X a*X a^2*X a*X  0 a^2*X a^2*X  X  0  0 a*X a*X a*X  0 a*X  X  0  X  X  X a^2*X a*X  0  0 a*X  0  0 a^2*X  X a*X  0 a^2*X a*X a*X  0 a^2*X  0 a*X a^2*X a^2*X  X a^2*X a^2*X a*X  0 a*X a^2*X  X  0 a*X a*X a^2*X  0  0  X  X a*X a*X a^2*X  0  X a^2*X
 0  0  0  X  0  X a^2*X  0  X a^2*X  X  0 a*X a^2*X  0 a^2*X  0  0 a*X a^2*X a*X  X  X  0 a^2*X a*X a*X  0 a*X  X  0 a*X a^2*X a*X a^2*X a*X  X a*X a*X a^2*X a^2*X  X a^2*X a*X a*X a*X a*X  X  X a*X a*X a*X  0  X  X a*X  0  0  X  X  0  X a^2*X  X  0 a^2*X  0  0  0 a*X a*X  0  0  X a*X  0  X  X
 0  0  0  0 a^2*X a^2*X  X a^2*X a*X  0 a^2*X  X  X a*X  X a*X  X a*X a^2*X a^2*X a^2*X a^2*X  0 a*X  0 a^2*X a^2*X  0 a*X a*X a^2*X  0 a*X  X  0  X  X  0  X  X  0  X a*X a*X a^2*X  X  0  0 a*X  0  0  0 a^2*X  X a^2*X  0 a^2*X a^2*X a*X a^2*X  X a*X a^2*X  X a*X a*X  0 a*X  X  X a*X a^2*X a^2*X  0  0  X a^2*X  X

generates a code of length 78 over F4[X]/(X^2) who´s minimum homogenous weight is 220.

Homogenous weight enumerator: w(x)=1x^0+717x^220+1743x^224+2796x^228+2640x^232+3600x^236+2691x^240+1578x^244+432x^248+72x^252+36x^256+36x^260+24x^268+9x^272+6x^276+3x^284

The gray image is a linear code over GF(4) with n=312, k=7 and d=220.
This code was found by Heurico 1.16 in 52.5 seconds.