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  1  1  1  1  1  1  2  1  1  1  1  1  1  1  1  1  2  X  1  1  1  2  1  1  1  X
 0  X  0 X^2+X+2  2 X^2+X  0  X X^2 X^2+X X^2+2  X X^2 X+2 X^2 X^2+X  0 X^2+X X^2 X+2  2 X^2+X+2  2 X^2+X  0 X+2 X^2 X+2 X^2 X+2 X^2 X^2+X+2 X^2+2 X^2+X+2  0 X^2+X+2 X^2+X+2  0  X X^2 X^2+X+2 X^2+2  X  2 X^2+X+2  0 X^2+X+2 X^2 X^2+2 X+2  0  X  0  2  X  X X^2+X  X X^2+X+2  X X^2  2 X^2 X^2+2 X^2+2  2  0 X^2+2 X^2  2 X^2+2 X^2 X^2+X  X X^2+X+2  X X^2+X X^2+X+2 X^2+X X+2  X X^2+X  X X^2+X+2  2
 0  0 X^2+2  0  0 X^2+2 X^2 X^2 X^2  2 X^2+2  2  2 X^2  2 X^2+2  0  0  2 X^2+2 X^2+2 X^2 X^2  2  2  0  0 X^2 X^2  2 X^2 X^2  0  0  2 X^2+2 X^2+2 X^2+2  2 X^2+2 X^2 X^2+2  0  2 X^2 X^2+2  2  0  0 X^2 X^2 X^2+2 X^2  0  0 X^2+2  0  2  2 X^2+2 X^2  2 X^2+2  0  2  0 X^2 X^2+2  0 X^2+2  2 X^2+2  2 X^2  0  0  2  2 X^2 X^2 X^2  0  2  0  0
 0  0  0 X^2+2 X^2 X^2+2 X^2  0  0  0 X^2 X^2+2 X^2 X^2+2  0  0  2  2 X^2+2 X^2  2  2 X^2+2 X^2 X^2 X^2+2  2  2  2  2 X^2 X^2+2 X^2+2 X^2  0  2 X^2 X^2  0  2 X^2  0  2  2  0 X^2+2 X^2+2 X^2  0 X^2  2 X^2+2  0 X^2+2 X^2  0  0 X^2  2  2 X^2+2 X^2+2 X^2+2  2  2  0 X^2+2  2 X^2+2 X^2 X^2+2  0  2  2  2 X^2+2  2 X^2 X^2  0  0 X^2+2  0  0 X^2

generates a code of length 85 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+32x^80+88x^81+137x^82+290x^83+406x^84+348x^85+278x^86+188x^87+82x^88+56x^89+73x^90+50x^91+14x^92+4x^93+1x^160

The gray image is a code over GF(2) with n=680, k=11 and d=320.
This code was found by Heurico 1.16 in 0.782 seconds.