Can you check the results of this chess match?

Question

As the sun broke the horizon I could feel the cold breeze on my face mixing with the warming city air. Like most mornings, I could hear Mr. and Mrs. Shamming discussing their afternoon plans during their daily jog. All the while, Mr. Punch (he got the nickname from fighting over card games) and Mr. Bim were trying to outplay each other in their daily chess matches.

I always enjoy watching them play so I sat at the nearby bench as usual, but I quickly noticed they were a bit grumpier than normal. "Pardon me, gentlemen, but what seems to be the issue?" I asked. Mr. Bim grunted, as was his standard reaction when ladies asked him questions. "It's a parody dear child.", Mr. Punch said softly. I stared at their chessboard as my mind melted into an enigmatic train of thought:

"You know Emma, if you stare much longer, I'm afraid the board may burst into flames and we won't be able to check the result of the game." said Mr. Punch. "You'd like that wouldn't ya, ya putz." snarled Mr. Bim.

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After Mr. Punch broke my concentration I caught a brief glimpse of their scoresheets, and neither had recorded Mr. Bim's last move; unfortunately, I can only remember the last two recorded moves:

Bxe4+, Rxe4+

To further complicate matters, Mr. Punch is tightly clutching the piece from Mr. Bim's last move.

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It would seem that Mr. Bim won the match this morning, but what was his final move? How does the state of the chessboard prove your answer?

Hints

These types of codes can also be seen as "self-correcting messages". The hidden message is broken and can be corrected to confirm Mr. Bim's last move was in fact the end of the game. Remove the parity bits and inspect what's left.

_{1: Note that the answer isn't as simple as identifying a valid play of checkmate. Instead, the entire chessboard is required to answer correctly. As such, additional tags include wordplay, computer-science, and knowledge.}

Answer 1

This puzzle is really about

error-correcting codes; more specifically, extended Hamming codes. (The name "Shamming" is a clue to this.) The way (one version of) these work is as follows. Pick some N (we shall take N=6). Our messages will be binary strings of length 2^N (here, 64, the number of squares on a chessboard). Now, for n=0,1,...,N-1, consider the subset S(n) of bits whose position in the message, written in binary, has a 1 in place n. So S(0) is the bits in places 1,3,...,63; S(1) is the bits in places 2,3,6,7,...,62,63; etc. We impose the restriction that each S(n) contains an even number of 1-bits, and we do that by allowing the sender of the message to choose all the bits except for the first one in each S(n), the bit in position 2^n, which we set to 0 or 1, whichever makes the parity of S(n) come out right. Oh, and we also don't let the sender choose the value of bit 0, which we set to make the whole message contain an even number of 1-bits. So we can send a message of length 2^n-n-1. And the nice thing is that all those constraints allow us to detect and correct any single-bit error (because if bit k is wrong then S(n) will come out with the wrong parity for exactly those n for which bit n of the number k is 1, so we can find k) and also to detect any two-bit error (because bit 0 will be "right" while at least one of the other parity bits is "wrong").

In this puzzle we'll

number the chessboard squares in "reading order" starting with 0 at top left and ending with 1 at bottom right, and take "no piece" to mean 0 and "piece" to mean 1.

We find

that parity bits 2 and 5 (in positions 4 and 32) are not what we expect, meaning that if there is at most one error then there is an error, in position 36, which is to say on e4. So it seems that Mr. Bim's last move was to capture the rook on e4. If this was his final move and delivered checkmate, that move was presumably with a diagonally-attacking piece, probably a queen since he already checked there with a bishop and had it taken. (Though of course he might at some point have underpromoted to a bishop.)

So

Mr. Bim's last move was Qxe4 mate.

Some other relevant features of the puzzle:

punched cards kinda represent binary data on a grid, just like we have here (I think the "card games" that gave him his nickname were not the usual sort); we are using parity check bits to check for single-bit errors. I already mentioned the significance of the name Shamming. Mr. Punch was presumably talking too quietly for Emma to realise that he said "parity", not "parody". I failed to figure out what Mr. Bim was a reference to, which is kinda embarrassing; as OP indicated in comments, of course it's a reference to IBM.

Now, what about the word we're supposed to extract? Well,

there is an actual message here (which I confess I initially hadn't thought plausible since OP managed to contrive a kinda-possible chess position). So what does it say? Well, if we remove the first row and first column (which are mostly made of parity bits) and restore the missing piece on e4 in line with the error correction, we get three empty rows followed by 1101101 1100001 1110100 1100101, which are the ASCII characters m, a, t, e. Very appropriate!