The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+1575x^220+3234x^224+3438x^228+2793x^232+2085x^236+1770x^240+1035x^244+420x^248+24x^252+3x^256+3x^260+3x^264

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 10 seconds.