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Linear code with complementary dual (LCD) [1]

Description

A linear code \(C\) admits a complementary dual if \(C\) and its dual code \(C^{\perp}\) do not share any nonzero codewords. Equivalently, \(\mathbb{F}_q^n = C \oplus C^{\perp}\).

Protection

Optimal binary and ternary LCD codes have been characterized [2]. LP bounds have been derived [3].

Rate

Asymptotically good LCD codes exist [1].

Decoding

The decoding problem reduces to finding the nearest codeword in \(C\) given a word in \(C^{\perp}\) [1].

Cousins

  • Group-algebra code— A group code \(C \leq \mathbb{F}_q G\) is LCD if and only if \(C=e \mathbb{F}_q G\) for an idempotent \(e\) satisfying \(e=\hat{e}\), and then \(C^{\perp}=(1-e)\mathbb{F}_q G\) [4; Thm. 16.7.6].
  • Cyclic linear \(q\)-ary code— A cyclic \([n,k]\) code with generator polynomial \(g(x)\) is LCD if and only if \(g(x)\) is self-reciprocal and \(\gcd( g(x), (x^{n}-1)/g(x) )=1\) [4; Cor. 16.7.11].
  • Reversible code— A reversible cyclic code is a cyclic code with self-reciprocal generator polynomial and is an LCD code [5; Thm. 2.10.3].
  • EA quantum LCD code— Asymptotically good maximal-entanglement EA Galois-qudit stabilizer codes can be constructed from LCD codes [6].
  • Maximal-entanglement EA Galois-qudit stabilizer code— Asymptotically good maximal-entanglement EA Galois-qudit stabilizer codes can be constructed from LCD codes [6].

Member of code lists

Primary Hierarchy

Classical Domain
Galois-field Kingdom
\(q\)-ary codeECC
Additive \(q\)-ary codeECC
Linear \(q\)-ary codeECC
Dual linear codeECC
Parents
Dual linear code
Linear code with complementary dual (LCD)

References

  1. [1]

    J. L. Massey, “Linear codes with complementary duals”, Discrete Mathematics 106–107, 337 (1992) DOI

  2. [2]

    M. Araya, M. Harada, and K. Saito, “Characterization and classification of optimal LCD codes”, (2021) arXiv:1908.03294

  3. [3]

    S. T. Dougherty, J.-L. Kim, B. Ozkaya, L. Sok, and P. Solé, “The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices”, (2015) arXiv:1506.01955

  4. [4]

    W. Willems, “Codes in Group Algebras”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 363-384 DOI

  5. [5]

    C. Ding, “Cyclic Codes over Finite Fields”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 45-60 DOI

  6. [6]

    K. Guenda, S. Jitman, and T. A. Gulliver, “Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes”, (2016) arXiv:1606.00134

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Zoo Code ID: lcd

Cite as:
“Linear code with complementary dual (LCD)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/lcd, arXiv:2606.11484
BibTeX:
@incollection{eczoo_lcd,
title={Linear code with complementary dual (LCD)},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/lcd}
}
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Permanent link:
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Cite as:

“Linear code with complementary dual (LCD)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/lcd, arXiv:2606.11484

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