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Poset code [1]

Root code for the Galois-field Kingdom

Description

Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(\mathbb{F}_q\), with its distance evaluated in the poset metric.

Protection

Poset codes are quantified with respect to the poset metric [2]. This metric is based on a partial ordering \(\leq\) on \([n]=\{1,2,\cdots,n\}\) and the notion of ideals generated by the support of a \(q\)-ary string. An ideal \(\langle supp(x) \rangle\) generated by \(x\in \mathbb{F}_q^n\) contains all elements of \([n]\) that are less than or equal to some element 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 support of \(x-y\), \(d_P(x,y) = |\langle supp(x-y) \rangle|\).

Generalizations of various bounds for ordinary \(q\)-ary codes have been developed for poset codes, including generalizations of MacWilliams identities [3,4]; see [2].

Notes

See book [5] for more details.

Cousins

  • Subspace code— Poset-code and subspace-code distance metric families intersect only at the Hamming metric [2].
  • \(t\)-design— Designs exist on ordered Hamming space [3,6].

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
Parents
Symmetric-space codeECC
Ordered Hamming space can be viewed as a finite symmetric space [3,6][7; Sec. 4.2.3][8; Table 3].
Poset code
Children
Niederreiter-Rosenbloom-Tsfasman (NRT) code

References

  1. [1]

    R. A. Brualdi, J. S. Graves, and K. M. Lawrence, “Codes with a poset metric”, Discrete Mathematics 147, 57 (1995) DOI

  2. [2]

    M. Firer, “Alternative Metrics”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 555-574 DOI

  3. [3]

    A. Barg and P. Purkayastha, “Bounds on ordered codes and orthogonal arrays”, (2009) arXiv:cs/0702033

  4. [4]

    S. Choi, J. Y. Hyun, H. K. Kim, and D. Y. Oh, “MacWilliams-type equivalence relations”, (2013) arXiv:1205.1090

  5. [5]

    M. Firer, M. M. S. Alves, J. A. Pinheiro, and L. Panek, Poset Codes: Partial Orders, Metrics and Coding Theory (Springer International Publishing, 2018) DOI

  6. [6]

    W. J. Martin and D. R. Stinson, “Association Schemes for Ordered Orthogonal Arrays and (T, M, S)-Nets”, Canadian Journal of Mathematics 51, 326 (1999) DOI

  7. [7]

    C. Bachoc, “Semidefinite programming, harmonic analysis and coding theory”, (2010) arXiv:0909.4767

  8. [8]

    C. Bachoc, D. C. Gijswijt, A. Schrijver, and F. Vallentin, “Invariant Semidefinite Programs”, International Series in Operations Research & Management Science 219 (2011) arXiv:1007.2905 DOI

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Zoo Code ID: poset

Cite as:
“Poset code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/poset, arXiv:2606.11484
BibTeX:
@incollection{eczoo_poset,
title={Poset code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/poset}
}
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Permanent link:
https://errorcorrectionzoo.org/c/poset

Cite as:

“Poset code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/poset, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/poset/poset.yml.