The generator matrix

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

generates a code of length 87 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 257.

Homogenous weight enumerator: w(x)=1x^0+144x^257+180x^260+480x^261+60x^264+144x^265+6x^268+6x^284+3x^288

The gray image is a linear code over GF(4) with n=348, k=5 and d=257.
This code was found by Heurico 1.16 in 0.128 seconds.