The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+109x^52+64x^53+64x^54+14x^56+1x^60+2x^68+1x^72

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 44.9 seconds.