The generator matrix

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

generates a code of length 77 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+12x^76+220x^77+12x^78+3x^80+2x^82+4x^85+2x^90

The gray image is a code over GF(2) with n=616, k=8 and d=304.
This code was found by Heurico 1.16 in 0.157 seconds.