The generator matrix

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

generates a code of length 88 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 257.

Homogenous weight enumerator: w(x)=1x^0+444x^257+588x^258+264x^259+6x^260+576x^261+552x^262+168x^263+21x^264+276x^265+204x^266+24x^267+24x^268+156x^269+168x^270+60x^271+84x^273+60x^274+12x^275+3x^276+108x^277+96x^278+12x^279+6x^280+36x^281+12x^282+36x^283+3x^284+48x^285+48x^286

The gray image is a linear code over GF(4) with n=352, k=6 and d=257.
This code was found by Heurico 1.16 in 32.2 seconds.