{
  "status": "shipping",
  "advantages": [],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "updated": "2026-10-15",
  "slug": "error-and-erasure-decoding",
  "term": "Error-and-erasure decoding",
  "aliases": [
    "Errors-and-erasures decoding",
    "Bounded-distance error-and-erasure decoding"
  ],
  "definition": "Error-and-erasure decoding reconstructs a codeword while accounting for both known erased positions and incorrect values at unknown positions.",
  "note": "For minimum distance d, a suitable bounded-distance decoder guarantees unique recovery with e errors among the non-erased positions and s erasures when 2e + s < d; for an MDS code this is 2e + s <= n - k. [forney] An erasure-only matrix solver need not locate unknown errors. Outside the guarantee, a single candidate can still be wrong, as the reproducible Hesela example demonstrates. [hesela-errors]",
  "limitations": [
    "The bound is a universal unique-decoding guarantee, not a failure probability or a guarantee that every out-of-bound pattern is unrecoverable.",
    "Decoder capability, trusted positions and codeword identity must be established separately; a code family name is insufficient."
  ],
  "related": [
    "symbol-erasure",
    "reed-solomon-code",
    "maximum-distance-separable-code",
    "codeword",
    "error-correcting-code"
  ],
  "citations": [
    {
      "id": "forney",
      "title": "G. David Forney. Principles of Digital Communication II, MIT 6.451 (2005), section 6.5.3: bounded-distance erasure-and-error correction.",
      "url": "https://ocw.mit.edu/courses/6-451-principles-of-digital-communication-ii-spring-2005/bb895c1dee9ce0b39d6846e0aa984981_MIT6_451S05_FullLecNotes.pdf",
      "accessed": "2026-10-15"
    },
    {
      "id": "senger",
      "title": "Senger, Sidorenko, Schober, Bossert and Zyablov. Adaptive Single-Trial Error/Erasure Decoding of Reed-Solomon Codes (2011), arXiv:1104.0576v1, sections II-V. Full five-page version reviewed.",
      "url": "https://arxiv.org/pdf/1104.0576v1",
      "accessed": "2026-10-15"
    },
    {
      "id": "hesela-errors",
      "title": "Hesela: exact GF(7) errors-and-erasures examples, including an ambiguous observation and false-identification negative control.",
      "url": "https://hesela.com/analyses/reed-solomon-matrix-recovery-guarantees/#errors-erasures",
      "accessed": "2026-10-15"
    }
  ]
}