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].

Notes

See [4; 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 [5].

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-geometric codes”, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280) DOI
[4]
F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
[5]
J. Fan et al., “Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically”, IEEE Transactions on Information Theory 1 (2022) DOI
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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}
}
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Permanent link:
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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/tree/main/codes/classical/q-ary_digits/ag/rs/alternant.yml.