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 a*X a^2*X a*X  X a*X a^2*X a*X  X  0 a*X  0 a*X  X  X a^2*X a*X  X a*X a*X a*X a^2*X  0 a*X a*X a^2*X a^2*X a*X  0  X  X  0 a*X  X a*X  X  0 a*X  0 a^2*X a^2*X a^2*X a*X a^2*X a^2*X  0  X  X a*X  X  X  0 a*X a^2*X a^2*X  X a^2*X  0  X a*X  0 a^2*X  X  X  0 a*X a^2*X a*X a^2*X  0  X a^2*X  X a*X a^2*X a*X  0  0
 0  0  X  0  0  0  0  X  X  X a*X  0 a*X a*X a*X a^2*X  0 a^2*X a^2*X  X  X a*X  0  X  0 a^2*X  X  0  X a*X a^2*X a*X  0 a^2*X  X  0 a*X  0  X  X a*X a*X a*X a*X a^2*X  X a*X  X a*X  0 a*X a*X  0  X  0  X  X a^2*X  0  X  X  0  0 a^2*X  X  X a*X a^2*X  0 a*X  X  0 a*X a*X  0 a*X a^2*X  0  X a*X  X  X  X a*X a^2*X  X  X  0
 0  0  0  X  0  0  X a^2*X a*X a*X a*X  0 a^2*X a*X a*X  X a^2*X a*X a*X  0 a^2*X  X a^2*X  X a*X  0  X  X  X  X  X  X  0  0  X a^2*X  0  X a*X a*X a*X a*X a^2*X a^2*X a*X  0  0  0  X  0 a*X a^2*X a^2*X a*X a^2*X  0  X  X  0  0 a*X  X a^2*X a^2*X  X a^2*X  0  X a*X  X  0 a*X a*X a^2*X  0 a*X  0  X  0 a*X a*X a^2*X  X a^2*X a*X  0 a*X a*X
 0  0  0  0  X  0 a^2*X  0  X a*X a^2*X a^2*X  X  0  0 a*X  0 a*X  0  X a*X  X  0 a*X  X  0 a*X a*X a^2*X a^2*X a^2*X a*X a*X a*X a*X a^2*X  X a*X  X a*X  0  X a^2*X a*X a*X a*X a^2*X  X a*X a^2*X a^2*X  X  X  0  X  X  X a^2*X a*X a^2*X a^2*X a^2*X  X a^2*X a*X a*X a^2*X  0  0  0  0  X a*X a^2*X a*X a^2*X a*X a*X  0  0  0 a^2*X a^2*X  X a*X a^2*X  0 a^2*X
 0  0  0  0  0  X  X  X a^2*X  X  X a*X  X  0 a*X a^2*X a^2*X a*X a*X a*X a*X a^2*X  X a*X a*X a*X a*X a^2*X a^2*X a*X a^2*X a*X  X a*X  0 a^2*X  0  0 a*X  X  X  0 a^2*X a^2*X  X  0 a^2*X a^2*X a^2*X  0 a*X  0  0 a^2*X  X a^2*X a*X a*X a^2*X  X a*X a*X a^2*X  0 a*X  X  0 a^2*X  X a^2*X a^2*X a*X a^2*X a^2*X a*X  X a*X a*X  0 a*X  0 a^2*X  0 a*X  X  X a*X  0

generates a code of length 88 over F4[X]/(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 4.23 seconds.