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

generates a code of length 84 over F4[X]/(X^2) who´s minimum homogenous weight is 244.

Homogenous weight enumerator: w(x)=1x^0+384x^244+2064x^248+576x^252+123x^256+192x^260+648x^264+96x^268+12x^272

The gray image is a linear code over GF(4) with n=336, k=6 and d=244.
This code was found by Heurico 1.16 in 0.307 seconds.