The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 41.

Homogenous weight enumerator: w(x)=1x^0+176x^41+489x^42+296x^43+335x^44+214x^45+264x^46+144x^47+69x^48+22x^49+18x^50+8x^51+2x^52+5x^54+4x^55+1x^56

The gray image is a linear code over GF(2) with n=352, k=11 and d=164.
This code was found by Heurico 1.16 in 0.422 seconds.