The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+32x^49+154x^50+220x^51+331x^52+496x^53+564x^54+638x^55+555x^56+412x^57+298x^58+182x^59+95x^60+42x^61+30x^62+12x^63+4x^64+8x^65+7x^66+2x^67+6x^68+2x^69+2x^70+2x^71+1x^82

The gray image is a code over GF(2) with n=440, k=12 and d=196.
This code was found by Heurico 1.16 in 0.453 seconds.