The generator matrix

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

generates a code of length 53 over Z2[X]/(X^4) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+106x^48+218x^49+349x^50+610x^51+434x^52+888x^53+381x^54+522x^55+241x^56+90x^57+85x^58+78x^59+49x^60+20x^61+15x^62+6x^63+2x^66+1x^84

The gray image is a linear code over GF(2) with n=424, k=12 and d=192.
This code was found by Heurico 1.16 in 0.328 seconds.