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  X  X  1  X  X  1  1  1
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^2 X^3 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^2  0 X^3 X^2  0 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2  0  0 X^3 X^3 X^3+X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^2 X^3+X^2 X^2  0  0 X^3 X^3 X^3+X^2  0  0 X^3+X^2 X^3  0 X^3
 0  0 X^3  0  0  0  0  0 X^3  0  0  0 X^3  0 X^3  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0 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 X^3  0  0  0 X^3  0  0  0 X^3  0  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0
 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  0  0  0  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3  0
 0  0  0  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3
 0  0  0  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0  0  0  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0  0 X^3 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+141x^44+40x^46+64x^47+81x^48+1408x^49+80x^50+64x^51+98x^52+8x^54+45x^56+17x^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 107 seconds.