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

generates a code of length 56 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 166.

Homogenous weight enumerator: w(x)=1x^0+132x^166+33x^168+48x^170+24x^172+3x^176+12x^182+3x^184

The gray image is a linear code over GF(4) with n=224, k=4 and d=166.
This code was found by Heurico 1.16 in 0.031 seconds.