The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 42.

Homogenous weight enumerator: w(x)=1x^0+292x^42+152x^43+244x^44+80x^45+167x^46+24x^47+58x^48+4x^50+1x^62+1x^64

The gray image is a linear code over GF(2) with n=352, k=10 and d=168.
This code was found by Heurico 1.16 in 1.08 seconds.