The generator matrix

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

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+152x^51+193x^52+480x^53+304x^54+734x^55+432x^56+728x^57+307x^58+430x^59+135x^60+128x^61+26x^62+26x^63+6x^64+8x^65+2x^67+2x^70+1x^72+1x^90

The gray image is a linear code over GF(2) with n=448, k=12 and d=204.
This code was found by Heurico 1.16 in 94.4 seconds.