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

generates a code of length 53 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 51.

Homogenous weight enumerator: w(x)=1x^0+144x^51+114x^52+88x^53+44x^54+32x^55+26x^56+56x^57+4x^58+2x^60+1x^64

The gray image is a code over GF(2) with n=424, k=9 and d=204.
This code was found by Heurico 1.16 in 0.141 seconds.