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

generates a code of length 68 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 192.

Homogenous weight enumerator: w(x)=1x^0+360x^192+144x^196+960x^200+1200x^204+1203x^208+78x^216+87x^224+42x^232+15x^240+6x^256

The gray image is a linear code over GF(4) with n=272, k=6 and d=192.
This code was found by Heurico 1.16 in 21.3 seconds.