The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+49x^44+10x^46+1x^48+2x^50+1x^52

The gray image is a linear code over GF(2) with n=352, k=6 and d=176.
This code was found by Heurico 1.16 in 0.047 seconds.