error-and-erasure-decoding · shipping · also Errors-and-erasures decoding, Bounded-distance error-and-erasure decoding

Error-and-erasure decoding

Error-and-erasure decoding reconstructs a codeword while accounting for both known erased positions and incorrect values at unknown positions.

Related Observatory article: Error-and-erasure decoding on hesela.com

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. [1] 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. [3]

Relations

related

symbol-erasure, reed-solomon-code, maximum-distance-separable-code, codeword, error-correcting-code

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.

Citations

1. G. David Forney. Principles of Digital Communication II, MIT 6.451 (2005), section 6.5.3: bounded-distance erasure-and-error correction. (accessed 2026-10-15)
2. 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. (accessed 2026-10-15)
3. Hesela: exact GF(7) errors-and-erasures examples, including an ambiguous observation and false-identification negative control. (accessed 2026-10-15)

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