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

generates a code of length 55 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 49.

Homogenous weight enumerator: w(x)=1x^0+58x^49+45x^50+32x^51+85x^52+110x^53+394x^54+648x^55+392x^56+98x^57+61x^58+16x^59+26x^60+42x^61+6x^62+8x^63+6x^64+12x^65+5x^66+1x^68+1x^74+1x^96

The gray image is a code over GF(2) with n=440, k=11 and d=196.
This code was found by Heurico 1.16 in 44.8 seconds.