The generator matrix

 1  0  0  0  1  1  1  1 3X X+2  1  1  1  X 2X  1 X+2 3X+2  0 2X+2 2X+2  1  1  1  1  1  1 3X  1 2X  1  1  X  1  1  1  1  X  1 2X+2  0 3X+2  X  1  1  2 2X+2  1  1 3X+2  1  1  1  1  1  X  1  1  1 2X 3X+2 2X+2  1 3X  1  1 3X  1  1  1  1  1  2  1  1  2  0 2X  1 3X  X  1 3X+2  1 2X+2  1  1  1
 0  1  0  0  0 2X+3 2X 3X+3  1  2  2  3 X+3  1  1 X+2  1  1  1  1 3X X+3  1 2X 3X+2  0 3X+3  X  3  1 3X X+2 3X X+1 X+1  0  X  1  1  1 X+2 3X  2  0 3X+1  1  1 X+1  0 2X 2X+2 X+2 2X 3X+3 X+1  1 2X+2  1  1  X 3X  2  3  1 2X+3 2X+3  X 3X 3X+2 X+2 2X 2X+1  1  0 X+1  2  1  1 X+1 3X+2  1 3X  X X+2  1 3X+1 2X+2 3X+2
 0  0  1  0  2 2X+2 2X+3  1 X+3  1 2X+1  3  X  X 3X+3  X  2  1 3X  3 2X+2 X+1  X 2X+2 2X+1 3X+1 X+3  1  X  2 3X+2 2X+2  1 3X+1  1 3X+2 X+1 3X+1 2X  1  1  1 3X+2 3X+1  2 3X+3  0  0 2X+2  1 3X+3 3X+2  1 3X+2 3X+1 X+3 X+2  1  X  1 2X  1  X  0 2X+1 X+3  1 2X+3 2X  1  0  0 X+3 3X  0  1 3X X+2  1  1 X+1 3X+1  1  2  0 3X 2X X+3
 0  0  0  1 X+3 3X+1 X+1 3X+3  X X+3 X+2 X+2 2X X+3  1 3X+2 3X+1 3X X+2  1  1 3X  0 2X+1 2X+1 2X X+1 X+2  1 3X  3  0 2X+3 2X+3 2X+2  0 3X+1  2  2 X+2 X+2 3X+1  1  X  X 3X+1 3X+3 2X+1 X+1 3X+2 2X+1 2X 2X X+1 2X+2 2X+3 3X+3  1 X+1 3X+1  1 2X  X 3X+2 X+3 2X+1  1 2X+3 3X+2 3X+2 3X X+3  0  3 3X+1 3X+3 2X+2 3X  X X+3 2X 3X+2 X+2  2 3X+1 3X+2 3X+1  3
 0  0  0  0 2X 2X 2X 2X  0  0 2X 2X 2X  0  0 2X  0  0  0  0  0 2X 2X 2X 2X 2X 2X  0 2X  0 2X 2X  0 2X 2X 2X 2X  0 2X  0 2X 2X 2X  0  0 2X 2X  0  0 2X  0  0  0  0  0 2X  0  0  0 2X 2X 2X  0 2X  0  0 2X  0  0  0  0  0  0  0  0  0 2X  0 2X  0 2X  0 2X  0 2X 2X 2X  0

generates a code of length 88 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+568x^80+1904x^81+3864x^82+5824x^83+7792x^84+10472x^85+13309x^86+14356x^87+15418x^88+14436x^89+12927x^90+10808x^91+7962x^92+5202x^93+3244x^94+1576x^95+671x^96+338x^97+241x^98+72x^99+34x^100+30x^101+14x^102+4x^103+2x^104+2x^105+1x^110

The gray image is a code over GF(2) with n=704, k=17 and d=320.
This code was found by Heurico 1.16 in 192 seconds.