Alternant code[1] 

Description

Given a length-\(n\) GRS code \(C\) over \(GF(q^m)\), an alternant code is the \(GF(q)\)-subfield subcode of the dual of \(C\).

Decoding

Variation of the Berlekamp-Welch algorithm [2].Guruswami-Sudan list decoder [3,4].

Notes

See [5; Ch. 12] for more details.

Parent

Cousin

  • Qubit CSS code — Alternant codes used in the CSS construction yield quantum codes that asymptotically achieve the quantum Gilbert-Varshamov bound [6].

References

[1]
H. J. Helgert, “Alternant codes”, Information and Control 26, 369 (1974) DOI
[2]
H. Helgert, “Decoding of alternant codes (Corresp.)”, IEEE Transactions on Information Theory 23, 513 (1977) DOI
[3]
V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometry codes”, IEEE Transactions on Information Theory 45, 1757 (1999) DOI
[4]
V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometric codes”, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280) DOI
[5]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[6]
J. Fan et al., “Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically”, IEEE Transactions on Information Theory 69, 262 (2023) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: alternant

Cite as:
“Alternant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/alternant
BibTeX:
@incollection{eczoo_alternant, title={Alternant code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/alternant} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/alternant

Cite as:

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

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