The generator matrix

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

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+44x^41+242x^42+172x^43+188x^44+118x^45+165x^46+32x^47+40x^48+18x^49+1x^50+1x^52+2x^56

The gray image is a linear code over GF(2) with n=352, k=10 and d=164.
This code was found by Heurico 1.16 in 0.39 seconds.