The generator matrix

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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 71.

Homogenous weight enumerator: w(x)=1x^0+96x^71+738x^72+1428x^73+2188x^74+2872x^75+3589x^76+3700x^77+4178x^78+3688x^79+3425x^80+2556x^81+1872x^82+1136x^83+686x^84+292x^85+158x^86+72x^87+28x^88+16x^89+18x^90+8x^91+13x^92+8x^93+2x^98

The gray image is a code over GF(2) with n=624, k=15 and d=284.
This code was found by Heurico 1.16 in 13.3 seconds.