// Numbas version: finer_feedback_settings {"name": "Hamming square code", "extensions": ["codewords"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["received_string", "errors", "decoded_words", "decoded_string", "received_words", "error_messages", "show_words", "words", "encwords", "num_words"], "name": "Hamming square code", "tags": [], "type": "question", "advice": "

Since we are using the Hamming square code, each code word is of length $9$. The message we received was $18$ bits long, so we have $2$ words to check and decode, $\\var{received_words[0]}$ and $\\var{received_words[1]}$.

Checking the received word $\\var{received_words[0]}$

We put this into the standard form for checking the square code - the received word is written in a $3 \\times 3$ grid, formed of the original word in the top-left $2 \\times 2$ square, with row and column sums on the sides.

\\[\\begin{array}{rr|c}\\var{received_words[0][0]}&\\var{received_words[0][1]}&\\var{received_words[0][2]}\\\\\\var{received_words[0][3]}&\\var{received_words[0][4]}&\\var{received_words[0][5]}\\\\ \\hline \\var{received_words[0][6]}&\\var{received_words[0][7]}&\\var{received_words[0][8]}\\end{array}\\]

{error_messages[errors[0]]}

We know that at most one error has occurred, so the decoded word is $\\var{decoded_words[0]}$.

Checking the received word $\\var{received_words[1]}$

As above, we put this into the standard form for checking the square code:

\\[\\begin{array}{rr|c}\\var{received_words[1][0]}&\\var{received_words[1][1]}&\\var{received_words[1][2]}\\\\\\var{received_words[1][3]}&\\var{received_words[1][4]}&\\var{received_words[1][5]}\\\\ \\hline \\var{received_words[1][6]}&\\var{received_words[1][7]}&\\var{received_words[1][8]}\\end{array}\\]

{error_messages[errors[1]]}

We know that at most one error has occurred, so the decoded word is $\\var{decoded_words[1]}$.

Putting the decoded words together, the decoded string is $\\var{latex(decoded_string)}$

", "rulesets": {}, "parts": [{"prompt": "

Assuming that the received message contains at most one error per codeword, correct any errors and then decode to find the original binary string.

", "marks": 1, "showCorrectAnswer": true, "displayAnswer": "{decoded_string}", "scripts": {}, "answer": "{decoded_string}", "type": "patternmatch"}], "statement": "

A binary string was encoded using Hamming's square code and then transmitted.

The string $\\var{latex(received_string)}$ was received.

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