[
  {
    "id": "binary-deletion-channel",
    "title": "Binary deletion channel",
    "abbreviation": "BDC",
    "status": "open",
    "category": "channels-with-memory",
    "featured": true,
    "updated": "2026-08-09",
    "summary": "Each input bit is independently deleted without an erasure marker. The exact capacity is unknown for every nontrivial deletion probability.",
    "tags": [
      "binary",
      "synchronization",
      "memory",
      "open-capacity"
    ],
    "parameters": [
      {
        "symbol": "d",
        "description": "Independent deletion probability.",
        "range": "0<d<1"
      }
    ],
    "model": {
      "input": "\\(X^n\\in\\{0,1\\}^n\\)",
      "output": "A random subsequence of the input, with no markers indicating deleted positions.",
      "law": "Each input bit is independently retained with probability \\(1-d\\), and the surviving bits keep their order.",
      "assumptions": [
        "Neither encoder nor decoder observes the deletion pattern.",
        "Average probability of error vanishes with blocklength."
      ]
    },
    "quantity": {
      "name": "Shannon capacity",
      "symbol": "C_{\\mathrm{del}}(d)",
      "units": "bits per input bit",
      "criterion": "Average-error operational capacity."
    },
    "capacity": {
      "kind": "bounds",
      "display": "0.1221(1-d)<C_{\\mathrm{del}}(d)\\le 0.3578(1-d)",
      "lower": "0.1221(1-d)",
      "upper": "0.3578(1-d)",
      "conditions": "The displayed pair is valid in the high-deletion regime \\(d\\ge0.64\\); stronger pointwise bounds may apply at individual \\(d\\).",
      "notes": "The site records simple uniform high-deletion bounds here. A future data extension should store the full pointwise numerical envelopes.",
      "conjectured": ""
    },
    "bounds": [
      {
        "id": "run-length-lower-2023",
        "direction": "lower",
        "relation": "C_{\\mathrm{del}}(d)>0.1221(1-d)",
        "method": "Optimized run-length input distributions in the Mitzenmacher-Drinea construction.",
        "year": 2023,
        "references": [
          "rubinstein-con-2023"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "parallel-blahut-arimoto-upper-2026",
        "direction": "upper",
        "relation": "C_{\\mathrm{del}}(d)\\le 0.3578(1-d)",
        "method": "Finite-block deletion-channel upper bounds computed by a parallelized Blahut-Arimoto algorithm.",
        "year": 2026,
        "conditions": "\\(d\\ge 0.64\\)",
        "references": [
          "pinto-ribeiro-2026"
        ],
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 2023,
        "event": "Rubinstein and Con improved both the high-deletion lower and upper constants.",
        "references": [
          "rubinstein-con-2023"
        ]
      },
      {
        "year": 2026,
        "event": "Pinto and Ribeiro improved the uniform high-deletion upper constant to 0.3578.",
        "references": [
          "pinto-ribeiro-2026"
        ]
      }
    ],
    "frontier": {
      "question": "Determine \\(C_{\\mathrm{del}}(d)\\), even at a single benchmark value such as \\(d=1/2\\), or substantially narrow the pointwise gap.",
      "bottleneck": "Deletions destroy symbol alignment, so finite-letter mutual-information optimizations grow exponentially and do not single-letterize.",
      "progress": [
        "Improve a certified pointwise lower or upper bound at a specified deletion probability.",
        "Prove structural properties of capacity-achieving stationary input processes.",
        "Replace large numerical optimizations by compact certificates that can ultimately be checked in Lean."
      ],
      "subproblems": [
        {
          "title": "Formalize the operational deletion-channel model and existence of capacity.",
          "status": "open",
          "url": ""
        },
        {
          "title": "Store machine-readable pointwise upper and lower envelopes with provenance.",
          "status": "open",
          "url": ""
        }
      ]
    },
    "formalization": {
      "status": "none",
      "notes": "The finite-channel core does not yet model variable-length outputs or synchronization errors.",
      "language": "",
      "files": []
    },
    "references": [
      "rubinstein-con-2023",
      "pinto-ribeiro-2026"
    ],
    "discussion": {},
    "url": "/problems/binary-deletion-channel/"
  },
  {
    "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/"
  },
  {
    "id": "general-relay-channel",
    "title": "General discrete memoryless relay channel",
    "status": "open",
    "category": "multi-user",
    "featured": true,
    "updated": "2026-08-09",
    "summary": "A relay assists communication from a source to a destination. Decode-forward and the cut-set bound do not coincide in general.",
    "tags": [
      "relay",
      "discrete-memoryless",
      "multi-user",
      "open-capacity"
    ],
    "model": {
      "input": "Source input \\(X\\) and relay input \\(X_r\\).",
      "output": "Relay observation \\(Y_r\\) and destination observation \\(Y\\).",
      "law": "A memoryless transition law \\(p(y,y_r\\mid x,x_r)\\).",
      "assumptions": [
        "The relay acts causally on its past observations.",
        "Average probability of error vanishes with blocklength."
      ]
    },
    "quantity": {
      "name": "Shannon capacity",
      "symbol": "C",
      "units": "bits per channel use",
      "criterion": "Average-error source-to-destination capacity."
    },
    "capacity": {
      "kind": "bounds",
      "display": "R_{\\mathrm{DF}}\\le C\\le R_{\\mathrm{cut}}",
      "lower": "R_{\\mathrm{DF}}=\\max_{p(x,x_r)}\\min\\{I(X;Y_r\\mid X_r),I(X,X_r;Y)\\}",
      "upper": "R_{\\mathrm{cut}}=\\max_{p(x,x_r)}\\min\\{I(X;Y,Y_r\\mid X_r),I(X,X_r;Y)\\}",
      "notes": "The bounds coincide for important subclasses, including degraded and reversely degraded relay channels, but not for the general model.",
      "conditions": "",
      "conjectured": ""
    },
    "bounds": [
      {
        "id": "decode-forward",
        "direction": "lower",
        "relation": "C\\ge R_{\\mathrm{DF}}",
        "method": "The relay decodes the source message and cooperatively forwards it.",
        "year": 1979,
        "references": [
          "cover-elgamal-1979"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "cut-set",
        "direction": "upper",
        "relation": "C\\le R_{\\mathrm{cut}}",
        "method": "Apply a converse across each cut separating source and destination.",
        "year": 1979,
        "references": [
          "cover-elgamal-1979"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1979,
        "event": "Cover and El Gamal established the principal coding theorems and capacity for major relay subclasses.",
        "references": [
          "cover-elgamal-1979"
        ]
      }
    ],
    "frontier": {
      "question": "Find a computable exact capacity characterization for the unrestricted discrete memoryless relay channel.",
      "bottleneck": "The relay can mix partial decoding, compression, and coherent cooperation across blocks, while standard cut-set arguments ignore the causal structure that limits these strategies.",
      "progress": [
        "Identify a new nontrivial subclass where an inner and outer bound coincide.",
        "Strengthen the converse using relay causality without introducing an intractable multiletter expression.",
        "Produce a channel family with a certified quantitative gap between standard bounds and a new scheme or converse."
      ],
      "subproblems": [
        {
          "title": "Formalize the causal relay code model and the classical cut-set statement.",
          "status": "open",
          "url": ""
        },
        {
          "title": "Curate exact-capacity subclasses as separate atlas entries.",
          "status": "open",
          "url": ""
        }
      ]
    },
    "formalization": {
      "status": "none",
      "notes": "A reusable causal network-code layer is needed before formalizing relay bounds.",
      "language": "",
      "files": []
    },
    "references": [
      "cover-elgamal-1979"
    ],
    "abbreviation": "",
    "parameters": [],
    "discussion": {},
    "url": "/problems/general-relay-channel/"
  },
  {
    "id": "additive-white-gaussian-noise-channel",
    "title": "Real additive white Gaussian noise channel",
    "abbreviation": "AWGN",
    "status": "solved",
    "category": "point-to-point",
    "featured": true,
    "updated": "2026-08-09",
    "summary": "The power-constrained real Gaussian channel has a closed-form capacity attained by a Gaussian input.",
    "tags": [
      "continuous-alphabet",
      "gaussian",
      "memoryless",
      "power-constraint"
    ],
    "parameters": [
      {
        "symbol": "P",
        "description": "Average input power constraint.",
        "range": "P \\ge 0"
      },
      {
        "symbol": "N",
        "description": "Noise variance per channel use.",
        "range": "N > 0"
      }
    ],
    "model": {
      "input": "\\(X \\in \\mathbb{R}\\)",
      "output": "\\(Y \\in \\mathbb{R}\\)",
      "law": "\\(Y=X+Z\\), where \\(Z\\sim\\mathcal{N}(0,N)\\) is independent across channel uses.",
      "assumptions": [
        "The encoder satisfies \\(\\mathbb{E}[X^2]\\le P\\).",
        "Average probability of error vanishes with blocklength."
      ]
    },
    "quantity": {
      "name": "Shannon capacity",
      "symbol": "C_{\\mathrm{AWGN}}(P,N)",
      "units": "bits per channel use",
      "criterion": "Average-error capacity without feedback."
    },
    "capacity": {
      "kind": "exact",
      "display": "C_{\\mathrm{AWGN}}(P,N)=\\frac12\\log_2\\!\\left(1+\\frac{P}{N}\\right)",
      "conditions": "\\(P\\ge 0\\) and \\(N>0\\).",
      "notes": "A zero-mean Gaussian input with variance P achieves capacity.",
      "lower": "",
      "upper": "",
      "conjectured": ""
    },
    "bounds": [
      {
        "id": "gaussian-achievability",
        "direction": "lower",
        "relation": "C\\ge \\frac12\\log_2(1+P/N)",
        "method": "Gaussian random coding and typical decoding.",
        "year": 1948,
        "references": [
          "shannon-1948"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "entropy-maximization-converse",
        "direction": "upper",
        "relation": "C\\le \\frac12\\log_2(1+P/N)",
        "method": "The Gaussian distribution maximizes differential entropy at fixed variance.",
        "year": 1948,
        "references": [
          "shannon-1948"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1948,
        "event": "Shannon identified the Gaussian-channel capacity formula and coding theorem.",
        "references": [
          "shannon-1948"
        ]
      }
    ],
    "formalization": {
      "status": "none",
      "notes": "The initial Lean library covers finite channels. Continuous probability and differential entropy are a later milestone.",
      "language": "",
      "files": []
    },
    "references": [
      "shannon-1948",
      "cover-thomas-2006"
    ],
    "frontier": null,
    "discussion": {},
    "url": "/problems/additive-white-gaussian-noise-channel/"
  },
  {
    "id": "sun-jafar-11-message-index-coding",
    "title": "Sun-Jafar 11-message multiple-unicast index-coding instance",
    "abbreviation": "SJ11",
    "status": "open",
    "category": "network-coding",
    "featured": true,
    "updated": "2026-08-09",
    "summary": "This is the smallest multiple-unicast instance identified by Sun and Jafar where Shannon inequalities do not give the best known converse.",
    "tags": [
      "index-coding",
      "multiple-unicast",
      "non-shannon",
      "nonlinear-coding",
      "open-capacity"
    ],
    "model": {
      "input": "Eleven independent messages \\(W_1\\) through \\(W_{11}\\) held by one broadcaster.",
      "output": "Receiver \\(j\\) must recover \\(W_j\\) from the common broadcast and its side information.",
      "law": "The unknown interferers at receivers \\(1\\) through \\(11\\) are respectively \\(\\{W_4,W_5\\}\\), \\(\\{W_5\\}\\), \\(\\{\\}\\), \\(\\{\\}\\), \\(\\{W_2\\}\\), \\(\\{W_2,W_3\\}\\), \\(\\{W_1,W_3\\}\\), \\(\\{W_2,W_4\\}\\), \\(\\{W_3,W_4\\}\\), \\(\\{W_3,W_5\\}\\), and \\(\\{W_4,W_6\\}\\). Every other nonrequested message is side information.",
      "assumptions": [
        "All messages have the same rate.",
        "Arbitrary blocklength and nonlinear codes are allowed.",
        "Each receiver has zero-error or asymptotically vanishing-error decoding under the stated capacity convention."
      ]
    },
    "quantity": {
      "name": "Nonlinear symmetric capacity",
      "symbol": "C_{\\mathrm{sym}}",
      "units": "message symbols per broadcast symbol",
      "criterion": "Symmetric multiple-unicast index-coding capacity."
    },
    "capacity": {
      "kind": "bounds",
      "display": "\\frac{5}{13}\\le C_{\\mathrm{sym}}\\le\\frac{11}{28}",
      "lower": "\\frac{5}{13}",
      "upper": "\\frac{11}{28}",
      "conjectured": "\\frac{5}{13}",
      "notes": "The linear symmetric capacity is exactly 5/13. Whether nonlinear coding improves it remains open.",
      "conditions": ""
    },
    "bounds": [
      {
        "id": "vector-linear-code",
        "direction": "lower",
        "relation": "C_{\\mathrm{sym}}\\ge\\frac{5}{13}",
        "method": "Explicit vector linear interference-alignment code.",
        "year": 2015,
        "references": [
          "sun-jafar-2015"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "zhang-yeung-converse",
        "direction": "upper",
        "relation": "C_{\\mathrm{sym}}\\le\\frac{11}{28}",
        "method": "Entropic converse using the Zhang-Yeung non-Shannon information inequality.",
        "year": 2015,
        "references": [
          "sun-jafar-2015",
          "zhang-yeung-1998"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "linear-capacity",
        "direction": "linear-exact",
        "relation": "C_{\\mathrm{sym}}^{\\mathrm{linear}}=\\frac{5}{13}",
        "method": "Achievability plus an Ingleton linear-rank converse.",
        "year": 2015,
        "references": [
          "sun-jafar-2015"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1998,
        "event": "Zhang and Yeung discovered the first non-Shannon information inequality used in the converse.",
        "references": [
          "zhang-yeung-1998"
        ]
      },
      {
        "year": 2015,
        "event": "Sun and Jafar established the 5/13 linear capacity and the 11/28 nonlinear outer bound.",
        "references": [
          "sun-jafar-2015"
        ]
      }
    ],
    "frontier": {
      "question": "Prove the nonlinear symmetric capacity is \\(5/13\\), or construct a nonlinear code with rate strictly above \\(5/13\\) and determine the resulting capacity.",
      "bottleneck": "Linear-rank inequalities close the linear problem, while known entropic inequalities have not reduced the nonlinear upper bound to the achievable rate.",
      "progress": [
        "Improve either endpoint with a fully specified code or auditable entropy certificate.",
        "Rule out a concrete family of nonlinear codes or bounded blocklengths.",
        "Formalize the existing 5/13 code, the Ingleton converse, or the 11/28 entropic converse in Lean."
      ],
      "subproblems": [
        {
          "title": "Formalize the exact 11-receiver instance from the interference sets.",
          "status": "done",
          "url": ""
        },
        {
          "title": "Formalize the explicit 5/13 vector linear code.",
          "status": "open",
          "url": ""
        },
        {
          "title": "Formalize the 11/28 Zhang-Yeung converse.",
          "status": "open",
          "url": ""
        }
      ]
    },
    "formalization": {
      "status": "definitions",
      "language": "Lean",
      "notes": "The shared multiple-unicast model and exact interference sets are formalized. No capacity bound is yet machine-checked.",
      "files": [
        {
          "path": "lean/CapacityAtlas/Network/IndexCoding.lean",
          "declaration": "CapacityAtlas.IndexCoding.sunJafar11",
          "description": "Exact 11-message instance encoded by its receiver interference sets."
        }
      ]
    },
    "references": [
      "sun-jafar-2015",
      "zhang-yeung-1998"
    ],
    "parameters": [],
    "discussion": {},
    "url": "/problems/sun-jafar-11-message-index-coding/"
  },
  {
    "id": "binary-erasure-channel",
    "title": "Binary erasure channel",
    "abbreviation": "BEC",
    "status": "solved",
    "category": "point-to-point",
    "updated": "2026-08-09",
    "summary": "Each transmitted bit is either received correctly or replaced by a visible erasure symbol.",
    "tags": [
      "binary",
      "discrete-memoryless",
      "symmetric",
      "erasure"
    ],
    "parameters": [
      {
        "symbol": "\\varepsilon",
        "description": "Erasure probability.",
        "range": "0\\le\\varepsilon\\le1"
      }
    ],
    "model": {
      "input": "\\(X\\in\\{0,1\\}\\)",
      "output": "\\(Y\\in\\{0,1,?\\}\\)",
      "law": "\\(Y=X\\) with probability \\(1-\\varepsilon\\), and \\(Y=?\\) with probability \\(\\varepsilon\\).",
      "assumptions": [
        "Channel uses are independent.",
        "Average probability of error vanishes with blocklength."
      ]
    },
    "quantity": {
      "name": "Shannon capacity",
      "symbol": "C_{\\mathrm{BEC}}(\\varepsilon)",
      "units": "bits per channel use",
      "criterion": "Average-error capacity."
    },
    "capacity": {
      "kind": "exact",
      "display": "C_{\\mathrm{BEC}}(\\varepsilon)=1-\\varepsilon",
      "conditions": "\\(0\\le\\varepsilon\\le1\\).",
      "lower": "",
      "upper": "",
      "conjectured": "",
      "notes": ""
    },
    "bounds": [
      {
        "id": "erasure-random-coding",
        "direction": "lower",
        "relation": "C\\ge1-\\varepsilon",
        "method": "Random coding, or linear erasure-correcting codes approaching the unerased fraction.",
        "year": 1948,
        "references": [
          "shannon-1948"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "revealed-erasure-converse",
        "direction": "upper",
        "relation": "C\\le1-\\varepsilon",
        "method": "At most one bit can pass through each unerased output position.",
        "year": 1948,
        "references": [
          "shannon-1948"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1948,
        "event": "The general discrete-memoryless coding theorem gives the formula by mutual-information maximization.",
        "references": [
          "shannon-1948"
        ]
      }
    ],
    "formalization": {
      "status": "definitions",
      "language": "Lean",
      "notes": "The transition kernel and its stochasticity proof are formalized. The operational capacity theorem is not yet formalized.",
      "files": [
        {
          "path": "lean/CapacityAtlas/Channels/Binary.lean",
          "declaration": "CapacityAtlas.Channel.binaryErasure",
          "description": "Binary erasure channel as a finite stochastic channel."
        }
      ]
    },
    "references": [
      "shannon-1948",
      "cover-thomas-2006"
    ],
    "featured": false,
    "frontier": null,
    "discussion": {},
    "url": "/problems/binary-erasure-channel/"
  },
  {
    "id": "binary-z-channel",
    "title": "Binary Z-channel",
    "abbreviation": "Z-channel",
    "status": "solved",
    "category": "point-to-point",
    "updated": "2026-08-09",
    "summary": "One binary symbol is transmitted perfectly while the other can flip in only one direction.",
    "tags": [
      "binary",
      "discrete-memoryless",
      "asymmetric"
    ],
    "parameters": [
      {
        "symbol": "p",
        "description": "Probability that input 1 is received as 0.",
        "range": "0\\le p\\le1"
      }
    ],
    "model": {
      "input": "\\(X\\in\\{0,1\\}\\)",
      "output": "\\(Y\\in\\{0,1\\}\\)",
      "law": "\\(P(Y=0\\mid X=0)=1\\) and \\(P(Y=0\\mid X=1)=p\\).",
      "assumptions": [
        "Channel uses are independent.",
        "Average probability of error vanishes with blocklength."
      ]
    },
    "quantity": {
      "name": "Shannon capacity",
      "symbol": "C_Z(p)",
      "units": "bits per channel use",
      "criterion": "Average-error capacity."
    },
    "capacity": {
      "kind": "exact",
      "display": "C_Z(p)=\\log_2\\!\\left(1+(1-p)p^{p/(1-p)}\\right)",
      "conditions": "\\(0<p<1\\), with the endpoint values \\(C_Z(0)=1\\) and \\(C_Z(1)=0\\) obtained by continuity.",
      "notes": "Unlike a symmetric channel, the capacity-achieving input is generally not uniform.",
      "lower": "",
      "upper": "",
      "conjectured": ""
    },
    "bounds": [
      {
        "id": "optimized-input-achievability",
        "direction": "lower",
        "relation": "C_Z(p)\\ge\\log_2\\!\\left(1+(1-p)p^{p/(1-p)}\\right)",
        "method": "Evaluate mutual information at the optimizing Bernoulli input distribution.",
        "year": 1948,
        "references": [
          "shannon-1948",
          "cover-thomas-2006"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "binary-input-converse",
        "direction": "upper",
        "relation": "C_Z(p)\\le\\log_2\\!\\left(1+(1-p)p^{p/(1-p)}\\right)",
        "method": "The discrete-memoryless channel converse followed by a one-variable maximization.",
        "year": 1948,
        "references": [
          "shannon-1948",
          "cover-thomas-2006"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "formalization": {
      "status": "definitions",
      "language": "Lean",
      "notes": "The asymmetric transition kernel is formalized. The closed-form mutual-information maximization is not.",
      "files": [
        {
          "path": "lean/CapacityAtlas/Channels/Binary.lean",
          "declaration": "CapacityAtlas.Channel.binaryZ",
          "description": "Binary Z-channel as a finite stochastic channel."
        }
      ]
    },
    "references": [
      "shannon-1948",
      "cover-thomas-2006"
    ],
    "featured": false,
    "timeline": [],
    "frontier": null,
    "discussion": {},
    "url": "/problems/binary-z-channel/"
  },
  {
    "id": "degraded-wiretap-channel",
    "title": "Degraded discrete memoryless wiretap channel",
    "status": "solved",
    "category": "security",
    "updated": "2026-08-09",
    "summary": "A sender communicates reliably to a legitimate receiver while keeping the message secret from a degraded eavesdropper.",
    "tags": [
      "secrecy",
      "discrete-memoryless",
      "degraded"
    ],
    "model": {
      "input": "\\(X\\in\\mathcal X\\)",
      "output": "Legitimate output \\(Y\\in\\mathcal Y\\) and eavesdropper output \\(Z\\in\\mathcal Z\\).",
      "law": "The channel is memoryless and physically degraded: \\(X\\to Y\\to Z\\).",
      "assumptions": [
        "Reliability is required at the legitimate receiver.",
        "Information leakage to the eavesdropper vanishes under the chosen secrecy criterion."
      ]
    },
    "quantity": {
      "name": "Secrecy capacity",
      "symbol": "C_s",
      "units": "secret bits per channel use",
      "criterion": "Reliable communication with asymptotically vanishing leakage."
    },
    "capacity": {
      "kind": "characterization",
      "display": "C_s=\\max_{P_X}\\bigl[I(X;Y)-I(X;Z)\\bigr]",
      "conditions": "The channel is physically degraded.",
      "lower": "",
      "upper": "",
      "conjectured": "",
      "notes": ""
    },
    "bounds": [
      {
        "id": "random-binning-achievability",
        "direction": "lower",
        "relation": "C_s\\ge\\max_{P_X}[I(X;Y)-I(X;Z)]",
        "method": "Stochastic encoding and random binning.",
        "year": 1975,
        "references": [
          "wyner-1975"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "degraded-secrecy-converse",
        "direction": "upper",
        "relation": "C_s\\le\\max_{P_X}[I(X;Y)-I(X;Z)]",
        "method": "Fano's inequality, secrecy, and degradedness.",
        "year": 1975,
        "references": [
          "wyner-1975"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1975,
        "event": "Wyner introduced the wiretap channel and proved the degraded secrecy-capacity formula.",
        "references": [
          "wyner-1975"
        ]
      }
    ],
    "formalization": {
      "status": "none",
      "notes": "Secrecy criteria and mutual-information converses are not yet present in the Lean core.",
      "language": "",
      "files": []
    },
    "references": [
      "wyner-1975",
      "cover-thomas-2006"
    ],
    "abbreviation": "",
    "featured": false,
    "parameters": [],
    "frontier": null,
    "discussion": {},
    "url": "/problems/degraded-wiretap-channel/"
  },
  {
    "id": "discrete-memoryless-channel",
    "title": "Finite discrete memoryless channel",
    "abbreviation": "DMC",
    "status": "solved",
    "category": "point-to-point",
    "updated": "2026-08-09",
    "summary": "The general finite memoryless point-to-point channel has a single-letter mutual-information characterization.",
    "tags": [
      "finite-alphabet",
      "discrete-memoryless",
      "foundational"
    ],
    "model": {
      "input": "\\(X\\in\\mathcal X\\), with finite \\(\\mathcal X\\).",
      "output": "\\(Y\\in\\mathcal Y\\), with finite \\(\\mathcal Y\\).",
      "law": "A stochastic matrix \\(W(y\\mid x)\\), used independently at each channel use.",
      "assumptions": [
        "The channel law is known to encoder and decoder.",
        "Average probability of error vanishes with blocklength."
      ]
    },
    "quantity": {
      "name": "Shannon capacity",
      "symbol": "C(W)",
      "units": "bits per channel use",
      "criterion": "Average-error capacity."
    },
    "capacity": {
      "kind": "characterization",
      "display": "C(W)=\\max_{P_X} I(X;Y)",
      "notes": "The maximum is over all probability distributions on the finite input alphabet.",
      "conditions": "",
      "lower": "",
      "upper": "",
      "conjectured": ""
    },
    "bounds": [
      {
        "id": "random-coding-achievability",
        "direction": "lower",
        "relation": "C(W)\\ge\\max_{P_X}I(X;Y)",
        "method": "Random coding and typical-set decoding.",
        "year": 1948,
        "references": [
          "shannon-1948"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "fano-converse",
        "direction": "upper",
        "relation": "C(W)\\le\\max_{P_X}I(X;Y)",
        "method": "Fano's inequality, the chain rule, and memorylessness.",
        "year": 1948,
        "references": [
          "shannon-1948"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1948,
        "event": "Shannon proved the noisy-channel coding theorem.",
        "references": [
          "shannon-1948"
        ]
      }
    ],
    "formalization": {
      "status": "definitions",
      "language": "Lean",
      "notes": "Shared finite stochastic channels, serial composition, and one-shot codes are formalized. The asymptotic coding theorem is not.",
      "files": [
        {
          "path": "lean/CapacityAtlas/FiniteChannel.lean",
          "declaration": "CapacityAtlas.FiniteChannel",
          "description": "Shared finite stochastic-channel structure."
        },
        {
          "path": "lean/CapacityAtlas/Code.lean",
          "declaration": "CapacityAtlas.OneShotCode",
          "description": "Shared one-shot encoder, decoder, and error definitions."
        }
      ]
    },
    "references": [
      "shannon-1948",
      "cover-thomas-2006"
    ],
    "featured": false,
    "parameters": [],
    "frontier": null,
    "discussion": {},
    "url": "/problems/discrete-memoryless-channel/"
  },
  {
    "id": "physically-degraded-broadcast-channel",
    "title": "Physically degraded two-receiver broadcast channel",
    "status": "solved",
    "category": "multi-user",
    "updated": "2026-08-09",
    "summary": "One transmitter sends private messages to two receivers whose outputs form a degradation chain.",
    "tags": [
      "broadcast",
      "discrete-memoryless",
      "degraded",
      "capacity-region"
    ],
    "model": {
      "input": "\\(X\\in\\mathcal X\\)",
      "output": "Receiver outputs \\(Y_1\\) and \\(Y_2\\).",
      "law": "A memoryless channel satisfying \\(X\\to Y_1\\to Y_2\\).",
      "assumptions": [
        "Receiver 1 is the stronger receiver.",
        "Each receiver requests its own private message."
      ]
    },
    "quantity": {
      "name": "Capacity region",
      "symbol": "\\mathcal C",
      "units": "rate pairs in bits per channel use",
      "criterion": "Vanishing average error at both receivers."
    },
    "capacity": {
      "kind": "region",
      "display": "\\mathcal C=\\bigcup_{p(u,x)}\\{(R_1,R_2):R_1\\le I(X;Y_1\\mid U),\\ R_2\\le I(U;Y_2)\\}",
      "conditions": "\\(U\\to X\\to Y_1\\to Y_2\\).",
      "lower": "",
      "upper": "",
      "conjectured": "",
      "notes": ""
    },
    "bounds": [
      {
        "id": "superposition-inner-region",
        "direction": "inner-region",
        "relation": "R_1\\le I(X;Y_1\\mid U),\\quad R_2\\le I(U;Y_2)",
        "method": "Superposition coding and successive decoding.",
        "year": 1973,
        "references": [
          "bergmans-1973"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "degraded-broadcast-outer-region",
        "direction": "outer-region",
        "relation": "R_1\\le I(X;Y_1\\mid U),\\quad R_2\\le I(U;Y_2)",
        "method": "Single-letter converse exploiting physical degradedness.",
        "year": 1973,
        "references": [
          "bergmans-1973"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1973,
        "event": "Bergmans proved the capacity region for degraded broadcast channels.",
        "references": [
          "bergmans-1973"
        ]
      }
    ],
    "formalization": {
      "status": "none",
      "notes": "Multi-user rate regions and auxiliary random variables are not yet formalized.",
      "language": "",
      "files": []
    },
    "references": [
      "bergmans-1973",
      "cover-thomas-2006"
    ],
    "abbreviation": "",
    "featured": false,
    "parameters": [],
    "frontier": null,
    "discussion": {},
    "url": "/problems/physically-degraded-broadcast-channel/"
  },
  {
    "id": "q-ary-symmetric-channel",
    "title": "q-ary symmetric channel",
    "abbreviation": "q-SC",
    "status": "solved",
    "category": "point-to-point",
    "updated": "2026-08-09",
    "summary": "The transmitted q-ary symbol is correct with probability 1-p and otherwise changes uniformly to one of the other symbols.",
    "tags": [
      "finite-alphabet",
      "discrete-memoryless",
      "symmetric"
    ],
    "parameters": [
      {
        "symbol": "q",
        "description": "Alphabet size.",
        "range": "q\\ge2"
      },
      {
        "symbol": "p",
        "description": "Total crossover probability.",
        "range": "0\\le p\\le(q-1)/q"
      }
    ],
    "model": {
      "input": "\\(X\\in\\{1,\\ldots,q\\}\\)",
      "output": "\\(Y\\in\\{1,\\ldots,q\\}\\)",
      "law": "\\(P(Y=X)=1-p\\); conditioned on an error, each of the \\(q-1\\) other symbols is equally likely.",
      "assumptions": [
        "Channel uses are independent.",
        "The displayed parameter range uses the usual monotone-noise convention."
      ]
    },
    "quantity": {
      "name": "Shannon capacity",
      "symbol": "C_q(p)",
      "units": "bits per channel use",
      "criterion": "Average-error capacity."
    },
    "capacity": {
      "kind": "exact",
      "display": "C_q(p)=\\log_2 q-h_2(p)-p\\log_2(q-1)",
      "conditions": "\\(q\\ge2\\) and \\(0\\le p\\le(q-1)/q\\).",
      "notes": "The uniform input distribution achieves capacity by channel symmetry.",
      "lower": "",
      "upper": "",
      "conjectured": ""
    },
    "bounds": [
      {
        "id": "uniform-qary-achievability",
        "direction": "lower",
        "relation": "C_q(p)\\ge\\log_2 q-h_2(p)-p\\log_2(q-1)",
        "method": "Uniform input and the discrete-memoryless coding theorem.",
        "year": 1948,
        "references": [
          "shannon-1948",
          "cover-thomas-2006"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "weakly-symmetric-converse",
        "direction": "upper",
        "relation": "C_q(p)\\le\\log_2 q-h_2(p)-p\\log_2(q-1)",
        "method": "Capacity formula for weakly symmetric channels.",
        "year": 1948,
        "references": [
          "shannon-1948",
          "cover-thomas-2006"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "formalization": {
      "status": "none",
      "notes": "A generic finite weakly symmetric channel and its capacity theorem are planned after the finite entropy layer.",
      "language": "",
      "files": []
    },
    "references": [
      "shannon-1948",
      "cover-thomas-2006"
    ],
    "featured": false,
    "timeline": [],
    "frontier": null,
    "discussion": {},
    "url": "/problems/q-ary-symmetric-channel/"
  },
  {
    "id": "two-user-discrete-memoryless-mac",
    "title": "Two-user discrete memoryless multiple-access channel",
    "abbreviation": "DM-MAC",
    "status": "solved",
    "category": "multi-user",
    "updated": "2026-08-09",
    "summary": "Two independent senders communicate to one receiver through a memoryless channel, yielding a polymatroidal capacity region.",
    "tags": [
      "multiple-access",
      "discrete-memoryless",
      "capacity-region"
    ],
    "model": {
      "input": "Independent encoder inputs \\(X_1\\in\\mathcal X_1\\) and \\(X_2\\in\\mathcal X_2\\).",
      "output": "A common receiver observes \\(Y\\in\\mathcal Y\\).",
      "law": "A memoryless transition law \\(p(y\\mid x_1,x_2)\\).",
      "assumptions": [
        "The two messages are independent.",
        "Time sharing is allowed."
      ]
    },
    "quantity": {
      "name": "Capacity region",
      "symbol": "\\mathcal C_{\\mathrm{MAC}}",
      "units": "rate pairs in bits per channel use",
      "criterion": "Vanishing average joint decoding error."
    },
    "capacity": {
      "kind": "region",
      "display": "\\bigcup_{p(q)p(x_1\\mid q)p(x_2\\mid q)}\\!\\left\\{\\begin{array}{l}R_1\\le I(X_1;Y\\mid X_2,Q),\\\\R_2\\le I(X_2;Y\\mid X_1,Q),\\\\R_1+R_2\\le I(X_1,X_2;Y\\mid Q)\\end{array}\\right\\}",
      "notes": "The auxiliary Q represents time sharing.",
      "conditions": "",
      "lower": "",
      "upper": "",
      "conjectured": ""
    },
    "bounds": [
      {
        "id": "mac-random-coding-inner",
        "direction": "inner-region",
        "relation": "R_1\\le I(X_1;Y\\mid X_2,Q), R_2\\le I(X_2;Y\\mid X_1,Q), R_1+R_2\\le I(X_1,X_2;Y\\mid Q)",
        "method": "Independent random codebooks and joint decoding.",
        "year": 1971,
        "references": [
          "ahlswede-1971",
          "liao-1972"
        ],
        "conditions": "",
        "notes": ""
      },
      {
        "id": "mac-converse-outer",
        "direction": "outer-region",
        "relation": "R_1\\le I(X_1;Y\\mid X_2,Q), R_2\\le I(X_2;Y\\mid X_1,Q), R_1+R_2\\le I(X_1,X_2;Y\\mid Q)",
        "method": "Fano's inequality and single-letterization with a time-sharing variable.",
        "year": 1972,
        "references": [
          "ahlswede-1971",
          "liao-1972"
        ],
        "conditions": "",
        "notes": ""
      }
    ],
    "timeline": [
      {
        "year": 1971,
        "event": "Ahlswede established the multiple-access coding theorem.",
        "references": [
          "ahlswede-1971"
        ]
      },
      {
        "year": 1972,
        "event": "Liao independently developed the multiple-access capacity region.",
        "references": [
          "liao-1972"
        ]
      }
    ],
    "formalization": {
      "status": "none",
      "notes": "Product input distributions, rate regions, and the MAC coding theorem are not yet formalized.",
      "language": "",
      "files": []
    },
    "references": [
      "ahlswede-1971",
      "liao-1972",
      "cover-thomas-2006"
    ],
    "featured": false,
    "parameters": [],
    "frontier": null,
    "discussion": {},
    "url": "/problems/two-user-discrete-memoryless-mac/"
  }
]