The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+163x^78+880x^79+1746x^80+2082x^81+3122x^82+3088x^83+3764x^84+3712x^85+3835x^86+3052x^87+2812x^88+1662x^89+1300x^90+680x^91+364x^92+224x^93+108x^94+72x^95+33x^96+32x^97+24x^98+4x^99+8x^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 15 seconds.