The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  0  X 2X+2  1  1  X  0  X 2X+2  1  X  1  X  1  1  1  1  X  X 2X  2  X  X 2X  2  1  1  1  1  1  1  1  1  2  X  X
 0  X 2X+2 X+2  0 X+2 2X+2 3X 2X 3X+2  2 3X 2X 3X+2  2  X X+2  X 3X  X  0 2X+2 X+2  X 3X  X  0 2X  2 2X+2 X+2 3X+2 3X 3X 3X+2  X  X  X 3X+2  X  X  X 2X 2X 2X+2  2 X+2 3X+2  X  X 2X+2  0  0
 0  0 2X 2X 2X  0  0 2X 2X 2X  0  0  0  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X 2X 2X  0 2X  0 2X  0 2X 2X  0 2X  0  0 2X  0 2X 2X  0  0 2X 2X  0 2X  0 2X 2X  0

generates a code of length 53 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 52.

Homogenous weight enumerator: w(x)=1x^0+96x^52+64x^53+80x^54+12x^56+3x^64

The gray image is a code over GF(2) with n=424, k=8 and d=208.
This code was found by Heurico 1.16 in 0.11 seconds.