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

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+72x^41+73x^42+140x^43+465x^44+138x^45+62x^46+56x^47+6x^48+10x^49+1x^86

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 0.047 seconds.