The generator matrix

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

generates a code of length 53 over Z2[X]/(X^4) who´s minimum homogenous weight is 49.

Homogenous weight enumerator: w(x)=1x^0+208x^49+502x^50+536x^51+648x^52+496x^53+572x^54+460x^55+365x^56+180x^57+60x^58+28x^59+17x^60+12x^61+8x^62+2x^66+1x^76

The gray image is a linear code over GF(2) with n=424, k=12 and d=196.
This code was found by Heurico 1.16 in 0.266 seconds.