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

generates a code of length 81 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 243.

Homogenous weight enumerator: w(x)=1x^0+192x^243+24x^244+36x^248+3x^256

The gray image is a linear code over GF(4) with n=324, k=4 and d=243.
This code was found by Heurico 1.16 in 0.296 seconds.