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  1  X  X  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  X  X  X  X  X
 0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  2  0  2  0  2  0  2  2  2  0  2  0  2  2  2  2  2  2  2  2  2  2  2  0  0  0  2  2  0  0  2  0  2  2  2  2  0  0  2  0  2  2  2  2  2  2  0  0  0  0  0
 0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  2  0  2  0  2  0  2  0  2  2  2  0  2  0  2  2  2  2  2  2  2  2  2  2  0  2  2  0  0  2  2  0  2  2  2  0  0  2  2  2  2  0  2  0  0  2  2  0  0  0  2  2
 0  0  0  2  0  0  0  0  0  0  0  2  2  2  2  2  2  2  2  2  2  2  0  0  2  2  2  2  0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2  0  2  2  0  0  2  0  2  0  2  2  2  0  2  0  0  0  2  0  2  0  2  2  0  0  2  2  0
 0  0  0  0  2  0  0  0  2  2  2  2  2  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  2  2  2  2  2  2  0  2  0  2  0  2  2  0  2  0  2  2  2  2  0  2  2  2  0  2  0  2  2  2  0  0  2  2  2  2  0  2  0  0  2  2  0  2
 0  0  0  0  0  2  0  2  2  2  0  0  0  0  2  2  2  2  0  0  0  0  0  0  2  2  2  2  2  2  0  0  0  0  0  2  0  2  2  0  0  2  2  0  2  2  0  2  2  2  2  0  2  2  0  2  2  0  2  2  2  2  0  0  2  0  2  2  0  0  0  0
 0  0  0  0  0  0  2  2  0  2  2  0  2  2  2  0  0  2  0  0  2  2  2  2  2  2  0  0  0  2  0  0  2  2  0  0  2  2  2  2  0  0  2  2  2  0  0  2  2  2  2  2  0  0  0  2  0  2  2  0  0  0  0  0  0  2  2  2  2  0  0  0

generates a code of length 72 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+56x^68+908x^72+56x^76+2x^80+1x^128

The gray image is a code over GF(2) with n=576, k=10 and d=272.
This code was found by Heurico 1.16 in 80.1 seconds.