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

generates a code of length 88 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 256.

Homogenous weight enumerator: w(x)=1x^0+63x^256+63x^260+822x^264+12x^268+27x^272+15x^276+12x^280+6x^284+3x^352

The gray image is a linear code over GF(4) with n=352, k=5 and d=256.
This code was found by Heurico 1.16 in 0.125 seconds.