Recognize a valid binary Golay codeword

Question

Are there any properties of a binary [24,12,8] Golay code which would allow me to say, for example, that a given 24-bit word is or is not a Golay codeword for some generator matrix? That is to say, is there any set of criterion which I could use in order to say "this 24-bit word is not a valid Golay codeword for any Golay generator matrix"?

Answer 1

If the word has weight 0, 8, 12, 16, or 24, then you can shuffle the columns of a generator matrix to get that word.

It likely wouldn't be one of the four Golay generators: Greedy, Icosahedron, Nonresidue, or Polynomial. But it doesn't have to be.

Answer 2

The general method for any linear code is the following: Call the given code $C$, $G$ its generator matrix and $c$ the given codeword. Now $c$ is a codeword of $C$ if and only if there is a vector $x$ such that $xG = c$. This is a linear equation on $x$, so the question can be answered efficiently by applying the standard method to solve a linear equation system (Gauss reduction).