The generator matrix

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

generates a code of length 85 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 243.

Homogenous weight enumerator: w(x)=1x^0+972x^243+390x^244+2220x^247+915x^248+2172x^251+747x^252+2328x^255+672x^256+1956x^259+621x^260+1200x^263+333x^264+792x^267+225x^268+444x^271+153x^272+204x^275+33x^276+6x^288

The gray image is a linear code over GF(4) with n=340, k=7 and d=243.
This code was found by Heurico 1.16 in 1.72 seconds.