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  X  X  X  X  X  X  X  X  X  X  X  X  X  2  X  X  X 2X  X  X  X  X  X  X
 0 2X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0 2X 2X 2X 2X 2X  0  0  0 2X  0 2X 2X 2X  0  0 2X  0  0  0
 0  0 2X  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X 2X  0 2X 2X  0  0 2X 2X 2X 2X  0  0  0  0  0  0  0 2X 2X 2X 2X 2X  0  0  0  0 2X  0 2X 2X  0  0  0  0  0 2X 2X  0
 0  0  0 2X  0  0  0 2X 2X 2X 2X 2X  0 2X 2X  0  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0 2X 2X  0  0  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0 2X 2X  0 2X 2X  0
 0  0  0  0 2X  0 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X  0  0 2X 2X 2X  0 2X  0  0  0  0  0  0  0 2X 2X
 0  0  0  0  0 2X 2X  0 2X 2X  0 2X 2X 2X  0  0 2X  0 2X 2X 2X  0  0 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0  0  0 2X 2X 2X  0 2X 2X 2X  0  0  0 2X 2X  0  0 2X  0 2X

generates a code of length 56 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+40x^52+128x^55+180x^56+128x^57+22x^60+11x^64+2x^76

The gray image is a code over GF(2) with n=448, k=9 and d=208.
This code was found by Heurico 1.16 in 0.093 seconds.