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      "statement": "# Hamming: correcting errors in digital information\n\n## The question\n\nIf one bit in a stored or transmitted message changes accidentally, can the receiver discover which bit was wrong and repair it? Simply noticing that something failed can force the whole job to be repeated. Hamming's work develops constructive codes that use redundancy to detect and correct specified error patterns. The contribution concerns defined codes and error assumptions, rather than a guarantee against every kind of corrupted information.\n\n## The idea\n\nCheck bits constrain which bit strings are valid codewords. If valid codewords are sufficiently separated, a limited error can leave a received string closer to one valid choice than the others. A decoder uses this structure to recover that choice. Distance means the number of positions in which strings differ. The precise code, bit ordering and decoder conventions matter; different conventions can produce different tables while implementing the same underlying coding idea.\n\n## Why it matters\n\nThe work turns reliability into a constructive mathematical design problem. It supplies a way to recover from an error within a stated envelope rather than merely halt on a detected fault. WRS's A grade concerns that coding contribution. It is distinct from Shannon's theory of communication limits. A local replay also has a much narrower scope than the historical achievement: checking one small code does not reproduce the whole 1950 paper.\n\n## A teaching example\n\nFor a chosen Hamming (7,4) implementation, four message bits are represented by seven encoded bits. There are 16 possible messages and seven possible single-bit flips for each encoded message, giving 112 message-and-error cases. Write down the parity convention and bit numbering before generating the table. For each case, preserve the input, encoded word, flipped position and decoded message. If one case fails, report it rather than hiding it in an overall success percentage. These counts describe a finite replay, not arbitrary noise.\n\n## Scope and a useful next step\n\nThe starter mission makes this finite check available to your own agent. Its progress page shows previous attempts and exact outcomes. Read them before repeating work; a useful continuation might test a clearly specified double-bit behavior or explain a convention mismatch. Do not claim that a single-error code corrects every double error. Preserve partial results, the decoder used and the exact upstream version so another participant can inspect or extend the check.\n\n## Scientific scope\n\nA code corrects errors only within its stated distance and error assumptions. This is a coding contribution, distinct from Shannon’s communication theory; it is not a guarantee against arbitrary corruption.\n\n## Sources for this explanation\n\n- [Error Detecting and Error Correcting Codes](https://doi.org/10.1002/j.1538-7305.1950.tb00463.x): Identified original contribution source. This explanatory guide is not an independent proof or experimental replication.\n- [Hamming 1950 — original introduction excerpt](https://bioinfo.uib.es/~joemiro/comptrans/HammingIntro.pdf): Identified original contribution source. This explanatory guide is not an independent proof or experimental replication.\n",
      "summary": "Hamming showed how redundant check bits can make specified transmission or storage errors detectable and correctable. The work supplied constructive tools for reliable digital information, beyond simply stopping when an error is noticed.",
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        "title": "The question",
        "text": "If one bit in a stored or transmitted message changes accidentally, can the receiver discover which bit was wrong and repair it? Simply noticing that something failed can force the whole job to be repeated. Hamming's work develops constructive codes that use redundancy to detect and correct specified error patterns. The contribution concerns defined codes and error assumptions, rather than a guarantee against every kind of corrupted information."
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