Hyperbolic evaluation code

Hyperbolic evaluation code[1–3]

Also known as Hyperbolic cascaded RS code.

Description

An evaluation code over polynomials in two variables. Generator matrices are determined in Ref. [3] following initial formulations of the codes as generalized concatenations of RS codes [1,2]; see [4; Ex. 4.26].

Protection

Minimum distance is determined by an order bound [3,5].

Parent

Polynomial evaluation code

Child

Reed-Muller (RM) code — RM codes are special cases of hyperbolic evaluation codes [3; Thm. 3 proof].

Cousin

Reed-Solomon (RS) code — Hyperbolic evaluation codes were initially formulated as generalized concatenations (a.k.a. cascades) of RS codes [1,2].

References

[1]: K. Saints and C. Heegard, “On hyperbolic cascaded Reed-Solomon codes”, Lecture Notes in Computer Science 291 (1993) DOI
[2]: Gui-Liang Feng and T. R. N. Rao, “Improved geometric Goppa codes. I. Basic theory”, IEEE Transactions on Information Theory 41, 1678 (1995) DOI
[3]: O. Geil and T. Høholdt, “On Hyperbolic Codes”, Lecture Notes in Computer Science 159 (2001) DOI
[4]: T. Høholdt, J.H. Van Lint, and R. Pellikaan, 1998. Algebraic geometry codes. Handbook of coding theory, 1 (Part 1), pp.871-961.
[5]: O. Geil and T. Hoholdt, “Footprints or generalized Bezout’s theorem”, IEEE Transactions on Information Theory 46, 635 (2000) DOI

Page edit log

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

Cite as:

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

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