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\([48,24,12]\) self-dual code

Description

An extended quadratic-residue code that is the unique self-dual doubly even \([48,24,12]\) code. It is extremal Type II, and its automorphism group is \(PSL(2,47)\) [2][1; Rem. 4.3.11].

Cousins

Member of code lists

Primary Hierarchy

Classical Domain
Binary Kingdom
Binary codeECC
Binary group-orbit codeECC
Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC
Parents
Linear binary code
Self-dual linear codeSelf-dual additive Linear \(q\)-ary ECC
The \([48,24,12]\) code is the unique self-dual doubly even code with those parameters [2][1; Rem. 4.3.11].
Divisible codeLinear \(q\)-ary ECC
The \([48,24,12]\) code is doubly even and hence Type II [1; Rem. 4.1.10]; it is the unique self-dual doubly even code with those parameters [2].
\([48,24,12]\) self-dual code

References

[1]
S. Bouyuklieva, “Self-dual codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[2]
S. K. Houghten, C. W. H. Lam, L. H. Thiel, and J. A. Parker, “The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code”, IEEE Transactions on Information Theory 49, 53 (2003) DOI
[3]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
[4]
M. Harada, A. Munemasa, and V. D. Tonchev, “A Characterization of Designs Related to an Extremal Doubly-Even Self-Dual Code of Length 48”, Annals of Combinatorics 9, 189 (2005) DOI
[5]
I. McLoughlin, “A group ring construction of the [48,24,12] type II linear block code”, Designs, Codes and Cryptography 63, 29 (2011) DOI
[6]
W. Willems, “Codes in Group Algebras.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[7]
S. P. Jain and V. V. Albert, “Transversal Clifford and T-Gate Codes of Short Length and High Distance”, IEEE Journal on Selected Areas in Information Theory 6, 127 (2025) arXiv:2408.12752 DOI
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Zoo Code ID: self_dual_48_24_12

Cite as:
“\([48,24,12]\) self-dual code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_48_24_12
BibTeX:
@incollection{eczoo_self_dual_48_24_12, title={\([48,24,12]\) self-dual code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/self_dual_48_24_12} }
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Permanent link:
https://errorcorrectionzoo.org/c/self_dual_48_24_12

Cite as:

“\([48,24,12]\) self-dual code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/self_dual_48_24_12

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/cyclic/quad_residue/self_dual_48_24_12.yml.