The generator matrix

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

generates a code of length 56 over Z2[X]/(X^4) who´s minimum homogenous weight is 51.

Homogenous weight enumerator: w(x)=1x^0+344x^51+971x^52+1712x^53+2101x^54+2254x^55+2280x^56+2020x^57+1880x^58+1272x^59+725x^60+462x^61+205x^62+110x^63+21x^64+10x^65+6x^66+4x^67+2x^68+2x^69+2x^73

The gray image is a linear code over GF(2) with n=448, k=14 and d=204.
This code was found by Heurico 1.16 in 4.55 seconds.