The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+768x^222+603x^224+2316x^226+819x^228+2436x^230+921x^232+1992x^234+534x^236+1980x^238+513x^240+1464x^242+381x^244+936x^246+219x^248+324x^250+87x^252+72x^254+15x^256+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 1.51 seconds.