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

generates a code of length 52 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 144.

Homogenous weight enumerator: w(x)=1x^0+285x^144+312x^152+3072x^156+210x^160+102x^168+66x^176+42x^184+3x^192+3x^208

The gray image is a linear code over GF(4) with n=208, k=6 and d=144.
This code was found by Heurico 1.16 in 42.3 seconds.