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

generates a code of length 86 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+221x^80+435x^82+32x^83+712x^84+224x^85+924x^86+224x^87+715x^88+32x^89+344x^90+145x^92+48x^94+15x^96+9x^98+14x^100+1x^148

The gray image is a code over GF(2) with n=688, k=12 and d=320.
This code was found by Heurico 1.16 in 1.47 seconds.