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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 codewords.

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].

Cousin

Member of code lists

Primary Hierarchy

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

References

[1]
J. L. Massey, “Linear codes with complementary duals”, Discrete Mathematics 106–107, 337 (1992) DOI
[2]
M. Araya, M. Harada, and K. Saito, “Characterization and classification of optimal LCD codes”, (2021) arXiv:1908.03294
[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
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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.), 2023. https://errorcorrectionzoo.org/c/lcd
BibTeX:
@incollection{eczoo_lcd, title={Linear code with complementary dual (LCD)}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, 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.), 2023. https://errorcorrectionzoo.org/c/lcd

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