The generator matrix

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

generates a code of length 85 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+162x^78+882x^79+1536x^80+2412x^81+2549x^82+3556x^83+3780x^84+4182x^85+3247x^86+3436x^87+2251x^88+1994x^89+1121x^90+792x^91+388x^92+202x^93+119x^94+66x^95+46x^96+10x^97+26x^98+4x^99+6x^100

The gray image is a code over GF(2) with n=680, k=15 and d=312.
This code was found by Heurico 1.16 in 13.5 seconds.