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

generates a code of length 72 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+24x^69+60x^70+104x^71+653x^72+104x^73+44x^74+24x^75+9x^76+1x^140

The gray image is a code over GF(2) with n=576, k=10 and d=276.
This code was found by Heurico 1.16 in 20.5 seconds.