The generator matrix

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

generates a code of length 54 over Z2[X]/(X^4) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+640x^48+1794x^49+3849x^50+5332x^51+7417x^52+8436x^53+10326x^54+8938x^55+8058x^56+4916x^57+3159x^58+1552x^59+749x^60+220x^61+88x^62+34x^63+15x^64+10x^65+2x^66

The gray image is a linear code over GF(2) with n=432, k=16 and d=192.
This code was found by Heurico 1.16 in 30.3 seconds.