The generator matrix

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

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+120x^49+176x^50+298x^51+355x^52+328x^53+263x^54+212x^55+107x^56+88x^57+40x^58+22x^59+16x^60+16x^61+1x^62+4x^63+1x^84

The gray image is a linear code over GF(2) with n=424, k=11 and d=196.
This code was found by Heurico 1.16 in 0.187 seconds.