The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X 1 1 X 1 X 1 X X X X^2 1 X 1 X X^2 X X^2 X X^3 0 X X^2 X^2 X X^3 X^2 1
0 X^3 0 X^3 0 X^3 0 X^3 0 X^3 0 X^3 0 X^3 0 X^3 X^3 0 X^3 X^3 0 X^3 X^3 0 X^3 0 X^3 0 0 X^3 X^3 X^3 0 0 X^3 X^3 X^3 0 0 X^3 0 0 0 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+1x^44+22x^45+2x^46+6x^47
The gray image is a linear code over GF(2) with n=352, k=5 and d=176.
This code was found by Heurico 1.16 in 0.032 seconds.