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].
Primary Hierarchy
Parents
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
- [4]
- W. Willems, “Codes in Group Algebras.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [5]
- C. Ding, “Cyclic Codes over Finite Fields.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [6]
- K. Guenda, S. Jitman, and T. A. Gulliver, “Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes”, (2016) arXiv:1606.00134
Page edit log
- Victor V. Albert (2023-07-18) — most recent
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.