The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X 0 1 X 0 1 1 1 X X^2 X X X X^2 1 1 1 1 1 1 1 1 1 X X X X X X X X X^2 X^2 0 0 X^2 0 0 X^2 X^2 0
0 X 0 X^2+X X^2 X^2+X X^2 X 0 X^2+X 0 X^2+X X^2 X X^2 X X^2+X X 0 X^2+X X X^2+X 0 X^2+X X X 0 X^2 X X X^2 X^2 X X 0 X^2 0 X^2 X^2+X X^2+X X^2+X 0 X^2 X X 0 X^2 0 X^2 X X X^2 X^2 0 X X X^2
0 0 X^2 X^2 X^2 0 0 X^2 0 0 X^2 X^2 X^2 X^2 0 0 0 X^2 0 X^2 0 0 X^2 X^2 X^2 0 X^2 X^2 0 X^2 X^2 0 X^2 0 0 0 X^2 X^2 0 0 X^2 0 0 X^2 0 X^2 X^2 X^2 X^2 0 X^2 0 X^2 X^2 X^2 0 0
generates a code of length 57 over Z2[X]/(X^3) who´s minimum homogenous weight is 57.
Homogenous weight enumerator: w(x)=1x^0+40x^57+6x^58+8x^59+6x^60+2x^62+1x^64
The gray image is a linear code over GF(2) with n=228, k=6 and d=114.
As d=114 is an upper bound for linear (228,6,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 6.
This code was found by Heurico 1.16 in 14 seconds.