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

generates a code of length 55 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 144.

Homogenous weight enumerator: w(x)=1x^0+189x^144+324x^148+438x^152+48x^153+417x^156+576x^157+420x^160+2592x^161+441x^164+5184x^165+420x^168+3888x^169+381x^172+369x^176+285x^180+180x^184+132x^188+57x^192+30x^196+6x^200+6x^204

The gray image is a linear code over GF(4) with n=220, k=7 and d=144.
This code was found by Heurico 1.16 in 2.41 seconds.