The generator matrix

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

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+414x^224+396x^226+1152x^227+660x^228+180x^230+222x^232+132x^236+138x^240+180x^242+384x^243+216x^244+12x^246+6x^248+3x^256

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 51.5 seconds.