The generator matrix

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

generates a code of length 57 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+14x^56+220x^57+14x^58+1x^64+4x^65+2x^66

The gray image is a code over GF(2) with n=456, k=8 and d=224.
This code was found by Heurico 1.16 in 0.141 seconds.