The generator matrix

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

generates a code of length 53 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 49.

Homogenous weight enumerator: w(x)=1x^0+96x^49+484x^50+718x^51+722x^52+656x^53+362x^54+348x^55+317x^56+136x^57+123x^58+86x^59+32x^60+8x^61+4x^62+3x^66

The gray image is a code over GF(2) with n=424, k=12 and d=196.
This code was found by Heurico 1.16 in 0.156 seconds.