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

generates a code of length 55 over Z2[X]/(X^4) 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 linear code over GF(2) with n=440, k=12 and d=196.
This code was found by Heurico 1.16 in 0.453 seconds.