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  1  1  1  1  1  X  1  X
 0  X  0  0  X  0  X  X (a+1)X (a+1)X (a+1)X (a+1)X (a+1)X (a+1)X  0  0  X  X  0  0  X  X (a+1)X (a+1)X  0  X aX (a+1)X  0  X aX (a+1)X  0  X aX (a+1)X  0  X (a+1)X  0  X (a+1)X  0  X (a+1)X  0  X (a+1)X  0  X (a+1)X aX aX aX aX aX aX aX aX aX aX aX aX aX  0  0  0  X  X  0  X  X  0 aX  X  X  X
 0  0  X  0 (a+1)X  X (a+1)X  0 (a+1)X (a+1)X  X  X (a+1)X  X  0  X (a+1)X  0  0  X (a+1)X  0 (a+1)X  X aX aX aX aX aX aX aX aX aX aX aX aX (a+1)X  X  0 (a+1)X  X  0 (a+1)X  X  0 aX aX  0 (a+1)X  X aX  0  0  0  X  X  X  0  X aX (a+1)X (a+1)X (a+1)X (a+1)X  0  0  X (a+1)X aX  X (a+1)X aX aX  0  0  X  0
 0  0  0  X  X  X aX (a+1)X  0  X  0  X aX aX aX (a+1)X  0 aX (a+1)X aX (a+1)X  X (a+1)X (a+1)X  0 (a+1)X  0 (a+1)X  X  X (a+1)X aX aX aX  X  0 (a+1)X  0  X aX (a+1)X (a+1)X  X  X aX (a+1)X  0  0  0 aX  X  0  X aX (a+1)X aX  X (a+1)X  0 aX (a+1)X aX  X  0  0  X (a+1)X  0  X aX  X  0  0  X aX (a+1)X (a+1)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+60x^224+48x^225+66x^228+288x^229+42x^232+432x^233+45x^236+33x^240+6x^244+3x^300

The gray image is a linear code over GF(4) with n=308, k=5 and d=224.
This code was found by Heurico 1.16 in 0.078 seconds.