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

generates a code of length 88 over F4[X,sigma]/(X^2) who´s minimum homogenous weight is 240.

Homogenous weight enumerator: w(x)=1x^0+147x^240+288x^244+417x^248+477x^252+192x^255+447x^256+1728x^259+306x^260+5184x^263+318x^264+5184x^267+330x^268+288x^272+207x^276+237x^280+225x^284+141x^288+126x^292+51x^296+30x^300+24x^304+18x^308+9x^312+6x^316+3x^340

The gray image is a linear code over GF(4) with n=352, k=7 and d=240.
This code was found by Heurico 1.16 in 4.37 seconds.