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  1  X  X  X  X  X  X  1  X  X  X  X  1  1  1  1  1  1  1  1  X  1
 0 X^3+X^2  0 X^2  0  0 X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^2 X^3 X^3 X^3+X^2 X^2 X^3 X^2 X^3+X^2  0 X^2 X^2  0 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^2  0 X^3 X^3  0  0 X^2 X^3+X^2 X^3  0 X^2 X^2 X^3+X^2 X^3 X^3+X^2 X^3
 0  0 X^3+X^2 X^2 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^2  0 X^2 X^2  0 X^3+X^2 X^3 X^3+X^2  0 X^3+X^2 X^2 X^3+X^2 X^2 X^3  0 X^3 X^3  0  0 X^3 X^2  0 X^2 X^3+X^2  0 X^3+X^2 X^3+X^2 X^3  0 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+9x^42+34x^43+173x^44+26x^45+6x^46+2x^47+2x^48+1x^54+2x^61

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.