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

generates a code of length 56 over Z2[X]/(X^3) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+120x^56+7x^64

The gray image is a linear code over GF(2) with n=224, k=7 and d=112.
As d=112 is an upper bound for linear (224,7,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 7.
This code was found by Heurico 1.16 in 0.0648 seconds.