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

generates a code of length 85 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+170x^79+65x^80+360x^81+186x^82+564x^83+274x^84+954x^85+260x^86+560x^87+179x^88+264x^89+26x^90+128x^91+21x^92+52x^93+8x^94+14x^95+2x^96+4x^99+1x^100+2x^101+1x^144

The gray image is a code over GF(2) with n=680, k=12 and d=316.
This code was found by Heurico 1.16 in 114 seconds.