The generator matrix

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

generates a code of length 87 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+502x^79+1950x^80+3530x^81+5773x^82+8444x^83+10069x^84+13138x^85+14167x^86+16068x^87+14741x^88+13354x^89+10024x^90+8114x^91+5059x^92+2884x^93+1700x^94+868x^95+384x^96+132x^97+107x^98+18x^99+20x^100+16x^101+5x^102+2x^103+2x^109

The gray image is a code over GF(2) with n=696, k=17 and d=316.
This code was found by Heurico 1.16 in 220 seconds.