{
  "slug": "varshamov-tenengolts-code",
  "term": "Varshamov-Tenengolts code",
  "aliases": [
    "VT code",
    "Binary VT code",
    "Varshamov-Tenengolts codes"
  ],
  "definition": "A binary Varshamov-Tenengolts code groups fixed-length bit strings by a weighted-sum congruence to correct one deleted or inserted bit.",
  "note": "For n bits and a fixed residue a, VT_a(n) contains strings satisfying sum(i*x_i) = a modulo n+1, with positions i numbered from 1. The sum uses ordinary integers, not XOR. The decoder must know n and a. Sloane describes the construction and single-deletion decoder. [sloane-vt]\n\nA deletion shortens the observation without marking its missing coordinate. An erasure preserves a known coordinate. These are different decoder inputs. Recovering the original string need not identify which identical bit in a run disappeared; racetrack read models illustrate this distinction. [chee-racetrack]\n\nThe binary condition does not directly extend to nucleotide values 0,1,2,3. Nguyen et al. give 213 and 132: both weighted sums are 13, yet either can lose a symbol to produce 13. Nonbinary VT constructions need additional or different constraints. [nguyen-vt]\n\nDecoding: Let w count received ones and let d = (a - sum(i*y_i)) modulo n+1 be the checksum deficiency. If d <= w, insert 0 with d ones to its right; otherwise insert 1 with d-w-1 zeros to its left. This requires a linear scan, not enumeration of the codebook. A returned word still assumes the correct block length, residue and one-deletion model. [sloane-vt]",
  "status": "research",
  "advantages": [],
  "limitations": [
    {
      "text": "One deletion OR one insertion is not a guarantee for a deletion plus an insertion, two deletions, or a substitution.",
      "citationIds": [
        "nguyen-vt"
      ]
    },
    {
      "text": "This mathematical family alone does not establish a complete DNA-storage codec, biochemical constraints or measured recovery rates.",
      "citationIds": [
        "nguyen-vt"
      ]
    }
  ],
  "related": [
    "error-correcting-code",
    "symbol-erasure",
    "racetrack-position-error",
    "dna-data-storage"
  ],
  "supersedes": [],
  "supersededBy": [],
  "firstSeen": null,
  "lastSeen": null,
  "citations": [
    {
      "id": "sloane-vt",
      "title": "N. J. A. Sloane. On Single-Deletion-Correcting Codes (revised 7 November 2002), sections 1-2, equation (4).",
      "url": "https://neilsloane.com/doc/dijen.pdf",
      "accessed": "2026-11-06"
    },
    {
      "id": "nguyen-vt",
      "title": "Nguyen, Cai and Siegel. Differential VT Codes and Differential Shifted VT Codes, arXiv:2311.04578v1 (2023), sections II-III.",
      "url": "https://arxiv.org/html/2311.04578v1",
      "accessed": "2026-11-06"
    },
    {
      "id": "chee-racetrack",
      "title": "Yeow Meng Chee et al. Coding for Racetrack Memories, arXiv:1701.06874v1 (2017), sections I-II.",
      "url": "https://arxiv.org/pdf/1701.06874v1",
      "accessed": "2026-11-06"
    }
  ],
  "updated": "2026-11-06",
  "provenance": [
    "Source-based mathematical definition. Status research describes the storage applications covered here, not the maturity of the theorem.",
    "Hesela's bounded binary enumeration and counterexamples are available at https://hesela.dev/research/vt-code/example.json with example.py and test_example.py; they are not media measurements or an all-length proof."
  ]
}