The generator matrix

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

generates a code of length 49 over Z2[X]/(X^4) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+152x^44+80x^46+64x^47+575x^48+384x^49+432x^50+64x^51+230x^52+63x^56+2x^60+1x^88

The gray image is a linear code over GF(2) with n=392, k=11 and d=176.
This code was found by Heurico 1.16 in 17 seconds.