The generator matrix

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

generates a code of length 55 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 53.

Homogenous weight enumerator: w(x)=1x^0+20x^53+66x^54+86x^55+55x^56+18x^57+5x^58+2x^59+2x^69+1x^74

The gray image is a code over GF(2) with n=440, k=8 and d=212.
This code was found by Heurico 1.16 in 0.094 seconds.