The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+342x^224+396x^226+1152x^227+768x^228+180x^230+186x^232+120x^236+198x^240+180x^242+384x^243+168x^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 0.243 seconds.