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

generates a code of length 57 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 169.

Homogenous weight enumerator: w(x)=1x^0+132x^169+45x^172+48x^173+15x^176+12x^185+3x^188

The gray image is a linear code over GF(4) with n=228, k=4 and d=169.
This code was found by Heurico 1.16 in 0.032 seconds.