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

generates a code of length 82 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+144x^77+667x^78+1160x^79+1034x^80+1160x^81+853x^82+810x^83+693x^84+510x^85+397x^86+286x^87+159x^88+186x^89+64x^90+32x^91+31x^92+2x^94+2x^96+1x^98

The gray image is a code over GF(2) with n=656, k=13 and d=308.
This code was found by Heurico 1.16 in 1.14 seconds.