# Norm-trace code[1]

## Description

Evaluation AG code of rational functions evaluated on points lying on a Miura-Kamiya curve in either affine or projective space. The family is named as such because the equations defining the curves can be expressed in terms of the field norm and field trace.

## Protection

Minimum distance is determined by an order bound [1].

## Parent

- Evaluation AG code — Norm-trace codes are evaluation AG codes with \(\cal X\) being a Miura-Kamiya curve [1].

## Child

- Hermitian code — Hermitian codes are evaluation AG codes with \(\cal X\) being a Hermitian curve [2][3; Ex. 2.74]. This curve is maximal, meaning that Hermitian codes are evaluation AG codes with maximum possible length given a fixed genus. They are a special case of norm-trace codes [1].

## References

- [1]
- O. Geil, “On codes from norm–trace curves”, Finite Fields and Their Applications 9, 351 (2003) DOI
- [2]
- R. E. Blahut, Algebraic Codes on Lines, Planes, and Curves (Cambridge University Press, 2001) DOI
- [3]
- T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.

## Page edit log

- Victor V. Albert (2024-08-17) — most recent

## Cite as:

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