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  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X  0 X^2+X X^3+X^2 X^3+X X^3  X X^3+X^2 X^3+X^2+X  X  0 X^3+X^2 X^2+X X^2+X X^3 X^2 X^3+X X^3+X  0 X^2+X X^2+X  0 X^3+X^2 X^3+X^2 X^3+X X^3+X X^3 X^3 X^3 X^2+X X^3+X^2 X^3+X^2+X X^3+X^2+X X^2 X^2 X^2+X X^3+X^2+X
 0  0 X^3+X^2  0 X^3+X^2 X^2  0 X^2 X^3 X^3 X^3 X^3 X^2 X^3+X^2 X^2 X^3+X^2 X^2  0  0  0 X^3+X^2 X^2 X^2 X^3 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^3 X^3+X^2 X^3 X^3+X^2 X^2 X^3  0 X^3  0  0 X^2
 0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0 X^3 X^3  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+88x^41+44x^42+172x^43+421x^44+164x^45+44x^46+84x^47+1x^48+4x^49+1x^84

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 0.594 seconds.