The generator matrix

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

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+28x^42+16x^43+167x^44+16x^45+20x^46+5x^48+1x^52+2x^60

The gray image is a linear code over GF(2) with n=352, k=8 and d=168.
This code was found by Heurico 1.16 in 0.047 seconds.