The generator matrix

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

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+308x^76+424x^78+272x^80+8x^82+8x^84+1x^88+1x^104+1x^112

The gray image is a code over GF(2) with n=624, k=10 and d=304.
This code was found by Heurico 1.16 in 0.656 seconds.