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

generates a code of length 55 over Z2[X]/(X^3) who´s minimum homogenous weight is 48.

Homogenous weight enumerator: w(x)=1x^0+56x^48+52x^50+128x^52+316x^54+240x^56+108x^58+60x^60+36x^62+23x^64+3x^68+1x^100

The gray image is a linear code over GF(2) with n=220, k=10 and d=96.
This code was found by Heurico 1.16 in 0.152 seconds.