Poset code

Poset code[1]

Description

Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), 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 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 [3,4]; 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

Classical codes

Primary Hierarchy

Parents

Homogeneous-space codeECC

Ordered Hamming space can be viewed as a finite symmetric space [3,6,7][8; Table 3].

Poset code

Children

Niederreiter-Rosenbloom-Tsfasman (NRT) code

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]: A. Barg and P. Purkayastha, “Bounds on ordered codes and orthogonal arrays”, (2009) arXiv:cs/0702033
[4]: S. Choi, J. Y. Hyun, H. K. Kim, and D. Y. Oh, “MacWilliams-type equivalence relations”, (2013) arXiv:1205.1090
[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]: 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]: C. Bachoc, “Semidefinite programming, harmonic analysis and coding theory”, (2010) arXiv:0909.4767
[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

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

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