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        "description": "The original work defines sparse parity-check ensembles and a specified iterative decoding procedure. Its channel, independence/tree-like neighborhood and limiting assumptions matter. Ensemble averages are not per-code finite-block guarantees, and cycles can affect iterative behavior. Later capacity-approaching irregular codes, decoding variants and communications deployments retain separate credit.",
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          "citation": "R.G.Gallager, Low-Density Parity-Check Codes, IRE Transactions on Information Theory IT-8(January1962),21–28",
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      "statement": "# Gallager: sparse parity checks and iterative error correction\n\nGallager's1962 article joins a regular sparse binary parity-check code family with local iterative decoding, separating ensemble-level optimal-decoding guarantees from the weaker conditions of its practical probabilistic decoder.\n\nA parity check asks whether a small group of bits has the expected even-or-odd total. Here each bit participates in only a fixed small number of checks. Instead of comparing a received block with every possible message, a decoder repeatedly combines nearby checks and channel probabilities. The article analyzes a random regular code ensemble and reports early finite computer experiments, while keeping limits caused by cycles and nonoptimal decoding explicit.\n\n## Scientific scope\n\nThe original work defines sparse parity-check ensembles and a specified iterative decoding procedure. Its channel, independence/tree-like neighborhood and limiting assumptions matter. Ensemble averages are not per-code finite-block guarantees, and cycles can affect iterative behavior. Later capacity-approaching irregular codes, decoding variants and communications deployments retain separate credit.\n\n## People and systems\n\nRobert G. Gallager\n\n\n\n## Original sources\n\n- [R.G.Gallager, Low-Density Parity-Check Codes, IRE Transactions on Information Theory IT-8(January1962),21–28](https://luca-giuzzi.unibs.it/Research/Coding/Files/Gallager.pdf)\n- [Gallager's MIT-hosted bibliography:1962 low-density parity-check article](https://web.mit.edu/gallager/www/pages/pubsold.html)\n",
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