# Poset code[1]

## Description

## Protection

Poset codes are quantified with respect to the poset metric [2]. This metric is based on a partial ordering \(\leq\) on subsets of \([n]=\{1,2,\cdots,n\}\) and the notion of ideals generated by the support of an element of a \(q\)-ary string. An ideal \(\langle \text{supp}(x) \rangle\) generated by \(x\in GF(q)^n\) contains all subsets of \([n]\) that are less than or equal to the subset in the support of \(x\) in the partial ordering. The poset metric between two strings \(x,y\) is then the cardinality of the ideal generated by the difference between the supports of \(x\) and \(y\), \(d_P(x,y) = |\langle \text{supp}(x-y) \rangle|\).

Generalizations of various bounds for ordinary \(q\)-ary codes have been developed for poset codes, including generalizations of MacWilliams identities [3]; see [2].

## Notes

## Parent

## Child

## Cousin

- Subspace code — Poset-code and subspace-code distance metric families intersect only at the Hamming metric [2].

## References

- [1]
- R. A. Brualdi, J. S. Graves, and K. M. Lawrence, “Codes with a poset metric”, Discrete Mathematics 147, 57 (1995) DOI
- [2]
- M. Firer, "Alternative Metrics." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [3]
- S. Choi et al., “MacWilliams-type equivalence relations”, (2013) arXiv:1205.1090
- [4]
- M. Firer et al., Poset Codes: Partial Orders, Metrics and Coding Theory (Springer International Publishing, 2018) DOI

## Page edit log

- Victor V. Albert (2024-09-11) — most recent

## Cite as:

“Poset code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/poset