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  X  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  X  1  1  1  1  1  1  1  1
 0  X  0  0  0  0  X  X  X aX  0  X (a+1)X  X (a+1)X (a+1)X (a+1)X aX  X (a+1)X  0 (a+1)X  0  0  X (a+1)X aX  X  0  0 aX  X aX  X  X (a+1)X  0 aX aX aX  0  0 aX aX aX  X (a+1)X aX  0  X  0 (a+1)X (a+1)X (a+1)X  X  0  X aX (a+1)X aX aX (a+1)X  X  X  0 aX aX  X (a+1)X  X  X aX (a+1)X  0  0  0 (a+1)X  0  0
 0  0  X  0  0  X (a+1)X aX aX aX  0  0 aX  X  0  0 (a+1)X  0 aX aX aX aX  X  X (a+1)X  X  X  0 (a+1)X aX  X  X  X  0  0 aX (a+1)X  X aX  0  X aX  0 aX aX  X aX aX (a+1)X  X (a+1)X (a+1)X aX (a+1)X (a+1)X  0  0 (a+1)X (a+1)X aX (a+1)X  0 (a+1)X (a+1)X aX (a+1)X (a+1)X aX (a+1)X (a+1)X aX aX aX (a+1)X (a+1)X aX (a+1)X (a+1)X  X
 0  0  0  X  0 (a+1)X  0  X aX (a+1)X  X  X  X  X  0 aX  0 aX  0 (a+1)X (a+1)X  X aX  0 (a+1)X aX (a+1)X aX (a+1)X  X (a+1)X (a+1)X aX  X aX aX (a+1)X aX (a+1)X  X aX  X  X  X (a+1)X  0 (a+1)X  X  X (a+1)X  0  X  0  0 (a+1)X  0 aX (a+1)X  X aX  0  0 (a+1)X  0 aX  X aX  0 (a+1)X  X  X  X  0  0  X  0  0 aX  0
 0  0  0  0  X  X  X (a+1)X  X  X  X aX  0 aX  X (a+1)X (a+1)X aX aX  0 aX (a+1)X aX aX (a+1)X  0 (a+1)X aX aX  0  X (a+1)X  0 (a+1)X (a+1)X (a+1)X  0  X (a+1)X  X  X (a+1)X (a+1)X  0 aX  0  X aX  0 aX aX aX  X  0  X (a+1)X  0 (a+1)X  X (a+1)X  X  0  0 (a+1)X aX  0 aX  0  0  0 (a+1)X  X  0  X aX (a+1)X aX  0  X

generates a code of length 79 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 224.

Homogenous weight enumerator: w(x)=1x^0+213x^224+270x^228+555x^232+1296x^236+1449x^240+120x^244+57x^248+51x^256+30x^260+9x^264+24x^272+12x^276+3x^280+3x^288+3x^304

The gray image is a linear code over GF(4) with n=316, k=6 and d=224.
This code was found by Heurico 1.16 in 28.4 seconds.