The generator matrix

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

generates a code of length 56 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+6x^54+30x^55+201x^56+9x^58+6x^60+2x^71+1x^82

The gray image is a code over GF(2) with n=448, k=8 and d=216.
This code was found by Heurico 1.16 in 0.11 seconds.