The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+72x^39+231x^40+274x^41+533x^42+632x^43+847x^44+504x^45+478x^46+174x^47+138x^48+86x^49+53x^50+48x^51+20x^52+2x^55+2x^56+1x^68

The gray image is a linear code over GF(2) with n=352, k=12 and d=156.
This code was found by Heurico 1.16 in 0.235 seconds.