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

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+120x^44+5x^48+2x^56

The gray image is a linear code over GF(2) with n=352, k=7 and d=176.
This code was found by Heurico 1.16 in 0.062 seconds.