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

generates a code of length 73 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+161x^68+80x^70+64x^71+567x^72+384x^73+432x^74+64x^75+224x^76+63x^80+7x^84+1x^136

The gray image is a code over GF(2) with n=584, k=11 and d=272.
This code was found by Heurico 1.16 in 27 seconds.