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

generates a code of length 83 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+597x^76+1670x^77+3234x^78+4324x^79+5809x^80+6198x^81+7348x^82+6986x^83+8316x^84+6476x^85+5430x^86+3902x^87+2498x^88+1286x^89+848x^90+304x^91+175x^92+38x^93+28x^94+34x^95+8x^96+8x^97+8x^98+2x^99+4x^100+4x^101

The gray image is a code over GF(2) with n=664, k=16 and d=304.
This code was found by Heurico 1.16 in 48.5 seconds.