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

generates a code of length 55 over Z2[X]/(X^3) who´s minimum homogenous weight is 44.

Homogenous weight enumerator: w(x)=1x^0+50x^44+118x^45+188x^46+292x^47+362x^48+452x^49+634x^50+810x^51+1147x^52+1432x^53+1742x^54+1930x^55+1725x^56+1556x^57+1131x^58+812x^59+617x^60+418x^61+325x^62+208x^63+172x^64+104x^65+71x^66+42x^67+22x^68+16x^69+4x^70+2x^71+1x^86

The gray image is a linear code over GF(2) with n=220, k=14 and d=88.
This code was found by Heurico 1.16 in 16.3 seconds.