The generator matrix

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

generates a code of length 77 over F4[X]/(X^2) who´s minimum homogenous weight is 224.

Homogenous weight enumerator: w(x)=1x^0+729x^224+468x^226+1020x^228+276x^230+447x^232+180x^234+180x^236+168x^238+240x^240+12x^242+204x^244+36x^246+108x^248+12x^250+12x^252+3x^280

The gray image is a linear code over GF(4) with n=308, k=6 and d=224.
This code was found by Heurico 1.16 in 0.141 seconds.