The generator matrix

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

generates a code of length 87 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 243.

Homogenous weight enumerator: w(x)=1x^0+420x^243+468x^244+264x^245+420x^246+1452x^247+1506x^248+708x^249+828x^250+3024x^251+2316x^252+1224x^253+1428x^254+3888x^255+3036x^256+1596x^257+1296x^258+4980x^259+3057x^260+1668x^261+1824x^262+4776x^263+3372x^264+1596x^265+1716x^266+4452x^267+2967x^268+1152x^269+996x^270+2976x^271+1794x^272+756x^273+552x^274+1272x^275+690x^276+204x^277+132x^278+348x^279+156x^280+48x^281+24x^282+60x^283+21x^284+21x^288+24x^292+9x^296+3x^300+6x^304+3x^308+3x^312+3x^316

The gray image is a linear code over GF(4) with n=348, k=8 and d=243.
This code was found by Heurico 1.16 in 28.6 seconds.