The generator matrix

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

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 71.

Homogenous weight enumerator: w(x)=1x^0+320x^71+1736x^72+2934x^73+4210x^74+5448x^75+7130x^76+7216x^77+8077x^78+7524x^79+6882x^80+5250x^81+3860x^82+2296x^83+1514x^84+564x^85+293x^86+172x^87+59x^88+20x^89+24x^90+6x^92

The gray image is a code over GF(2) with n=624, k=16 and d=284.
This code was found by Heurico 1.16 in 50.6 seconds.