# Glynn code[1]

## Description

The unique trace-Hermitian self-dual \([10,5,6]_9\) code, constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve.

The Glynn code is the unique trace-Hermitian code for its parameters, and is not Euclidean self-dual [2–4].

## Parents

- Projective geometry code — The Glynn code is constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve.
- Maximum distance separable (MDS) code — The Glynn code is a rare example of an MDS code that is not related to an RS code.

## Cousins

- Reed-Solomon (RS) code — The only other inequivalent \([10,5,6]_9\) code is an RS code, which is the unique Euclidean self-dual code for its parameters, and which is not Hermitian self-dual [2–4].
- \([[9,1,5]]_3\) quantum Glynn code — Applying the Hermitian construction to the Glynn code yields a \([[10,0,6]]_3\) state [5,6]. The \([[9,1,5]]_3\) quantum Glynn code can be obtained by applying the Hermitian construction to the shortened Glynn code [5; Corr. 4] (cf. [7; Exam. 7]).

## References

- [1]
- D. G. Glynn, “The non-classical 10-arc of PG(4, 9)”, Discrete Mathematics 59, 43 (1986) DOI
- [2]
- T. Baicheva, I. Bouyukliev, S. Dodunekov, and W. Willems, “On the [10; 5; 6] Reed-Solomon and Glynn codes,” Mathematica Balkanica, New Series, vol. 18, pp. 67–78, 200
- [3]
- T. A. Gulliver, J.-L. Kim, and Y. Lee, “New MDS or Near-MDS Self-Dual Codes”, IEEE Transactions on Information Theory 54, 4354 (2008) DOI
- [4]
- M. Grassl and T. A. Gulliver, “On circulant self-dual codes over small fields”, Designs, Codes and Cryptography 52, 57 (2009) DOI
- [5]
- M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) (2015) arXiv:1502.05267 DOI
- [6]
- S. Ball, “Some constructions of quantum MDS codes”, (2021) arXiv:1907.04391
- [7]
- S. Ball and R. Vilar, “The geometry of Hermitian self-orthogonal codes”, (2021) arXiv:2108.08088

## Page edit log

- Victor V. Albert (2022-08-11) — most recent

## Cite as:

“Glynn code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/glynn