Alternant code

Alternant code [1–4]

Description

Given a length-\(n\) GRS code \(C\) over \(\mathbb{F}_{q^m}\), an alternant code is the \(\mathbb{F}_q\)-subfield subcode of the dual of \(C\); see [5; Ch. 12]. Its parity-check matrix is an alternant matrix.

Decoding

Variation of the Berlekamp-Welch algorithm [6].Euclidean algorithm; see [5; Ch. 12] for more details.Guruswami-Sudan list decoder [7,8].

Notes

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

Cousins

Generalized RS (GRS) code— Alternant codes are subfield subcodes of GRS codes [4].
Berlekamp code— Berlekamp codes reduce to narrow-sense alternant codes for \(p=2\) [9; Ch. 10.6].
Qubit CSS code— Alternant codes used in the CSS construction yield quantum codes that asymptotically achieve the quantum GV bound [10].

Member of code lists

Primary Hierarchy

Classical Domain

Galois-field Kingdom

\(q\)-ary codeECC

Additive \(q\)-ary codeECC

Linear \(q\)-ary codeECC

Parents

Linear \(q\)-ary code

Alternant code

Children

Goppa code

Goppa codes are a special case of alternant codes [5; Ch. 12].

Chien-Choy generalized BCH (GBCH) code

GBCH codes are a special case of alternant codes [5; Ch. 12].

Generalized Srivastava code

Generalized Srivastava codes are a special case of alternant codes [5; Ch. 12].

References

[1]
H. J. Helgert, “Noncyclic generalizations of BCH and srivastava codes”, Information and Control 21, 280 (1972) DOI
[2]
H. J. Helgert, “Alternant codes”, Information and Control 26, 369 (1974) DOI
[3]
H. J. Helgert, “Binary primitive alternant codes”, Information and Control 27, 101 (1975) DOI
[4]
P. Delsarte, “On subfield subcodes of modified Reed-Solomon codes (Corresp.)”, IEEE Transactions on Information Theory 21, 575 (1975) DOI
[5]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
[6]
H. Helgert, “Decoding of alternant codes (Corresp.)”, IEEE Transactions on Information Theory 23, 513 (1977) DOI
[7]
V. Guruswami and M. Sudan, “Improved decoding of Reed-Solomon and algebraic-geometry codes”, IEEE Transactions on Information Theory 45, 1757 (1999) DOI
[8]
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) 28 DOI
[9]
R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
[10]
J. Fan, J. Li, Y. Wang, Y. Li, M.-H. Hsieh, and J. Du, “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

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2022-01-02)
Khalil Guy (2022-01-02)
Victor V. Albert (2022-03-24)
Manasi Shingane (2021-12-05)

Cite as:

“Alternant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/alternant, arXiv:2606.11484

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