The generator matrix

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

generates a code of length 78 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 226.

Homogenous weight enumerator: w(x)=1x^0+336x^226+264x^227+66x^228+936x^230+888x^231+78x^232+216x^234+24x^236+336x^238+63x^240+192x^242+120x^243+6x^244+264x^246+264x^247+15x^248+24x^250+3x^264

The gray image is a linear code over GF(4) with n=312, k=6 and d=226.
This code was found by Heurico 1.16 in 0.27 seconds.