I was surprised the article didn't add another couple of sentences explaining the finite field construction of the Hamming code. It seems like the important property of β is generating the finite field of size 8, so that its powers from 1 to 7 are distinct.
Then every 7-bit sequence gets assigned one of the 8 values in the field via the polynomial. The 0s are the code elements, as it says, and every other sequence is a power of β indicating which bit to flip to get a code element.
Then every 7-bit sequence gets assigned one of the 8 values in the field via the polynomial. The 0s are the code elements, as it says, and every other sequence is a power of β indicating which bit to flip to get a code element.