Alternant code

Alternant code[1]

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

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

See [4; Ch. 12] for more details.

Calderbank-Shor-Steane (CSS) stabilizer code — Alternant codes used in the CSS construction yield quantum codes that asymptotically achieve the quantum Gilbert-Varshamov bound [5].

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