The generator matrix

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

generates a code of length 80 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+44x^76+194x^78+128x^79+323x^80+128x^81+132x^82+60x^84+10x^86+3x^88+1x^136

The gray image is a code over GF(2) with n=640, k=10 and d=304.
This code was found by Heurico 1.16 in 0.422 seconds.