The generator matrix

 1  0  0  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1 a^2*X  1  0  1  X  1 a*X  1  1  1  1 a^2*X  1  1  1 a*X  1  1  1  1  1  1  1  1  1  1  0  1 a^2*X  1  1  1  1  1  1  1  1  X  X  1  0  X  1  1  1 a*X  1  X  1  0  1  1 a*X  1  1  1  1 a^2*X  1 a^2*X  1  0  1  1  1  1  1  1  1  1
 0  1  0 a^2*X a*X  X  1 a^2*X+a a^2 a^2*X+1 a*X+1 X+1 a*X+a  1 X+a^2 X+a a^2*X+a^2  a  1 a*X+a^2  1  a  1  0  1 a*X+1 a^2*X X+a^2 a^2  1 X+1 a^2*X+a  X  0 a*X+a^2 a^2*X+a X+a a*X+1 a^2 a^2*X a*X+a^2  1 a*X+a X+a^2  1 a^2*X+1  1 a*X+a a*X a*X+a^2 X+a  0  1 a^2*X+a a^2*X+a^2  1  1  a  1  1 a^2*X+1 X+1 a*X a*X a^2*X a*X  X  1 X+a^2 a^2*X+1  1 a^2 a*X+a^2 X+a a*X+1  1 a^2  1 a*X+a  1 X+a^2 a*X+a a*X X+1 a*X+1 a^2  1 X+a
 0  0  1  1  a a^2 X+a^2 a^2*X+a^2 a*X+a^2 X+a a*X+1  X X+1 a^2*X+a^2 a*X+a a^2*X+a a^2*X+1  0 a*X+1 a*X X+a  X a^2*X+a a*X a^2*X+1 X+a^2 a^2*X+1 a*X+1 a*X+a  X X+a a^2*X+a a^2*X+a^2  1 X+a^2  X  1  0 a^2*X+1 a^2 a^2*X+a a*X+a X+a  a  1 a*X+1 a^2*X a^2*X+a^2 a^2*X  a a^2*X  1 a^2 X+a a^2*X a^2 a*X+a a^2 a^2*X+1  0 a^2*X+a a*X+a^2 a^2*X+a^2  1 a*X  1 a*X+1  X X+1  1 a*X+a^2 a*X+1 a^2*X  0 a^2*X+1 a^2*X+a^2 a*X  1 X+a^2 a*X+a a^2*X+a^2  1 X+a^2 a^2*X a^2*X+a^2 X+a^2  X X+1

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

Homogenous weight enumerator: w(x)=1x^0+396x^257+600x^258+264x^259+6x^260+648x^261+552x^262+180x^263+24x^264+228x^265+240x^266+24x^267+21x^268+216x^269+60x^270+84x^273+108x^274+60x^275+3x^276+48x^277+96x^278+36x^279+3x^280+60x^281+36x^282+12x^283+6x^284+48x^285+36x^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 4.45 seconds.