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

generates a code of length 53 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 144.

Homogenous weight enumerator: w(x)=1x^0+72x^144+165x^148+48x^150+192x^152+432x^154+144x^156+1296x^158+129x^160+1296x^162+66x^164+57x^168+66x^172+54x^176+21x^180+24x^184+18x^188+12x^192+3x^200

The gray image is a linear code over GF(4) with n=212, k=6 and d=144.
This code was found by Heurico 1.16 in 0.163 seconds.