Remark. I have zero knowledge of any fancy terminology used to describe hamming codes, and am only aware of the basic principle of how they work (and why it works), that given a message, a few extra bits can be used as parity checks on the original message to locate and correct up to 1 error. (i.e. I've seen the 3Blue1Brown videos on the topic)
The idea of parity checking seems so ingenious that when I search for codes that can correct up to 2 errors (or more), I get complicated looking codes, eg. Reed Solomon, something something generating polynomial, CRC yadda yadda.
I wonder, are there any such codes that only rely on parity checks (perhaps one will need a lot more such checks) to detect and correct a 2 bit error in a message? Is such a code efficient at doing so?
For instance, given a $11$-bit message, a minimum of $4$ redundancy bits would be required to correct up to 1 error. What is the minimum number of extra redundancy bits required to correct upto a 2-bit error? And what would the associated parity checks look like? Is this as efficient as the original hamming code, for whatever metric of "efficiency" means? Why is such a code not popular? i.e. I searched alot on google with keywords eg. "2 bit error correction", "parity checks", "hamming code 2 bit error", with no results.