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 X^2  1  X X^2  1  1  X  X  X  1
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^2 X^3 X^3+X^2 X^3+X^2  0 X^3+X^2  0 X^3 X^2  0 X^3 X^3+X^2 X^2 X^3+X^2 X^2  0 X^3  0 X^3 X^3+X^2 X^2  0 X^3 X^3  0 X^3+X^2 X^3+X^2 X^2 X^2  0  0 X^3 X^3+X^2 X^3+X^2  0 X^2 X^3 X^2  0  0 X^3+X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0
 0  0 X^3  0  0  0  0  0 X^3  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0 X^3  0  0  0 X^3 X^3  0
 0  0  0 X^3  0  0  0 X^3  0  0  0  0 X^3  0  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3
 0  0  0  0 X^3  0  0  0  0  0  0 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3 X^3
 0  0  0  0  0 X^3  0 X^3  0  0 X^3  0 X^3  0  0  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3  0  0  0
 0  0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0  0 X^3  0  0 X^3  0 X^3  0  0 X^3  0 X^3  0 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+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 linear code over GF(2) with n=440, k=11 and d=196.
This code was found by Heurico 1.16 in 44.9 seconds.