{
  "id": "binary-symmetric-channel",
  "title": "Binary symmetric channel",
  "abbreviation": "BSC",
  "status": "solved",
  "category": "point-to-point",
  "featured": true,
  "updated": "2026-08-09",
  "summary": "Each bit is independently flipped with probability p. This is the canonical finite noisy channel.",
  "tags": [
    "binary",
    "discrete-memoryless",
    "symmetric"
  ],
  "parameters": [
    {
      "symbol": "p",
      "description": "Crossover probability.",
      "range": "0\\le p\\le1/2"
    }
  ],
  "model": {
    "input": "\\(X\\in\\{0,1\\}\\)",
    "output": "\\(Y\\in\\{0,1\\}\\)",
    "law": "\\(Y=X\\oplus Z\\), where \\(Z\\sim\\mathrm{Bernoulli}(p)\\) independently across uses.",
    "assumptions": [
      "The range p at most 1/2 removes the equivalent relabelled regime.",
      "Average probability of error vanishes with blocklength."
    ]
  },
  "quantity": {
    "name": "Shannon capacity",
    "symbol": "C_{\\mathrm{BSC}}(p)",
    "units": "bits per channel use",
    "criterion": "Average-error capacity."
  },
  "capacity": {
    "kind": "exact",
    "display": "C_{\\mathrm{BSC}}(p)=1-h_2(p)",
    "conditions": "\\(0\\le p\\le1/2\\), where \\(h_2\\) is binary entropy.",
    "notes": "The uniform input distribution achieves capacity.",
    "lower": "",
    "upper": "",
    "conjectured": ""
  },
  "bounds": [
    {
      "id": "uniform-input-achievability",
      "direction": "lower",
      "relation": "C\\ge1-h_2(p)",
      "method": "Apply the noisy-channel coding theorem to independent uniform inputs.",
      "year": 1948,
      "references": [
        "shannon-1948"
      ],
      "conditions": "",
      "notes": ""
    },
    {
      "id": "symmetric-channel-converse",
      "direction": "upper",
      "relation": "C\\le1-h_2(p)",
      "method": "Maximize I(X;Y) over the binary input distribution; symmetry makes the uniform input optimal.",
      "year": 1948,
      "references": [
        "shannon-1948"
      ],
      "conditions": "",
      "notes": ""
    }
  ],
  "timeline": [
    {
      "year": 1948,
      "event": "Shannon's noisy-channel coding theorem determines the capacity.",
      "references": [
        "shannon-1948"
      ]
    }
  ],
  "formalization": {
    "status": "definitions",
    "language": "Lean",
    "notes": "The transition kernel and its zero-noise identity are formalized. The coding theorem and entropy optimization remain open.",
    "files": [
      {
        "path": "lean/CapacityAtlas/Channels/Binary.lean",
        "declaration": "CapacityAtlas.Channel.binarySymmetric",
        "description": "Binary symmetric channel as a finite stochastic channel."
      },
      {
        "path": "lean/CapacityAtlas/Channels/Binary.lean",
        "declaration": "CapacityAtlas.Channel.binarySymmetric_zero",
        "description": "At p=0 the BSC is the identity channel."
      }
    ]
  },
  "references": [
    "shannon-1948",
    "cover-thomas-2006"
  ],
  "frontier": null,
  "discussion": {},
  "url": "/problems/binary-symmetric-channel/"
}