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

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+67x^80+240x^81+271x^82+338x^83+444x^84+560x^85+421x^86+536x^87+397x^88+256x^89+219x^90+182x^91+54x^92+28x^93+23x^94+24x^95+20x^96+4x^97+2x^98+8x^99+1x^144

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 1.08 seconds.