The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  X  X  X  X  X  1  1  X  1  1  X  X  X  X  1  X  1  1  1  1  1  1  1  X  X  X  X  1  1  1  1  1  1  1  1 X^2
 0 X^2+2  0 X^2  0  0 X^2 X^2+2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  0  2 X^2  0 X^2 X^2+2 X^2 X^2 X^2+2  2  0  2  0 X^2 X^2 X^2+2 X^2+2 X^2 X^2  2  2  2  0  0 X^2+2 X^2+2 X^2+2 X^2+2  0  0  2  2  0  2 X^2 X^2+2  0  0 X^2+2 X^2+2  2
 0  0 X^2+2 X^2  2 X^2 X^2+2  2  2 X^2 X^2+2  2  0 X^2+2 X^2  0  0 X^2 X^2 X^2+2  2  0 X^2 X^2+2  2 X^2  2 X^2+2 X^2 X^2+2  0 X^2+2 X^2  2  0  2 X^2+2  0 X^2+2 X^2 X^2+2 X^2  0  2  0  2  2  0  0 X^2 X^2  0  2 X^2+2 X^2+2  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+76x^56+128x^57+32x^58+16x^60+2x^72+1x^80

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 0.094 seconds.