The generator matrix

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

generates a code of length 83 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+186x^78+658x^79+1086x^80+1112x^81+1069x^82+934x^83+752x^84+692x^85+507x^86+446x^87+294x^88+196x^89+131x^90+38x^91+58x^92+16x^93+9x^94+4x^95+1x^98+1x^104+1x^106

The gray image is a code over GF(2) with n=664, k=13 and d=312.
This code was found by Heurico 1.16 in 1.17 seconds.