The generator matrix

 1  0  0  1  1  1 X^3  1  1  1  1 X^2  0 X^3+X^2+X  1  1 X^3+X  1  X  1 X^3+X^2+X  1  X X^2+X  1 X^3+X  1  0  1 X^3+X^2  1 X^2 X^3+X^2  1  1  1 X^2+X  1  X  X X^3+X^2  1  X  1 X^2+X  1  1  1  1  1 X^2 X^2+X  1
 0  1  0 X^3 X^2+1 X^3+X^2+1  1  X X^3+X X^3+X^2+X+1 X^2+X+1  1  1 X^2+X  1 X^3+X^2  1 X^2+X+1  0 X^3+X  1 X^3+X^2+X+1  1  1  1  X X^2+X X^3+X^2+X X^3+X^2+X  1 X^3+X^2 X^3  1 X+1 X^3+X^2 X^3  1  1  1 X^3+X^2+X X^2+X X^3+X+1 X^3+X^2 X^3+X^2  1 X^2 X^3+X X^3 X^2+X X^2+1 X^3+X^2 X^3+X^2  0
 0  0  1 X^3+X+1 X+1 X^3 X^3+X+1 X^3+X X^3+1  1  X  1 X^3+X  1 X^3+X^2+1 X^2+X X^2 X^2  1 X^2+X+1 X^3+X+1 X+1  1 X^2+X X^3+X^2+X  1  0  1 X^3+X+1 X^2+X+1 X^3+1  1 X^3 X^2 X+1 X^2+X  0 X^3+X^2 X^3+X^2+X+1  1  1 X^3+X^2+1  1 X^2+X+1  1 X^2+1 X^3+X^2+1 X^2+1  1 X^3+X  1  1  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+70x^49+583x^50+662x^51+634x^52+662x^53+482x^54+310x^55+275x^56+168x^57+153x^58+36x^59+41x^60+12x^61+5x^62+1x^68+1x^70

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 0.672 seconds.