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

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

Homogenous weight enumerator: w(x)=1x^0+83x^80+252x^81+229x^82+310x^83+432x^84+566x^85+527x^86+564x^87+370x^88+252x^89+153x^90+134x^91+56x^92+74x^93+40x^94+12x^95+18x^96+8x^97+10x^98+4x^99+1x^142

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 1.09 seconds.