The generator matrix

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

generates a code of length 44 over Z2[X]/(X^4) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+107x^38+606x^39+1570x^40+2732x^41+3487x^42+5372x^43+5055x^44+5704x^45+3419x^46+2476x^47+1299x^48+552x^49+235x^50+72x^51+51x^52+20x^53+6x^54+2x^55+2x^58

The gray image is a linear code over GF(2) with n=352, k=15 and d=152.
This code was found by Heurico 1.16 in 8.13 seconds.