The generator matrix

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

generates a code of length 80 over F4[X]/(X^2) who´s minimum homogenous weight is 224.

Homogenous weight enumerator: w(x)=1x^0+81x^224+174x^228+48x^231+171x^232+432x^235+129x^236+1296x^239+123x^240+1296x^243+84x^244+48x^248+57x^252+30x^256+30x^260+27x^264+30x^268+12x^272+15x^276+6x^284+3x^288+3x^308

The gray image is a linear code over GF(4) with n=320, k=6 and d=224.
This code was found by Heurico 1.16 in 0.308 seconds.