## Description

An extended quadratic-residue code that is known to be the only self-dual doubly even code at its parameters [1].

The code's automorphism group is \(PSL_2(47)\) [2; Remark 4.3.11].

## Parents

- Binary quadratic-residue (QR) code
- Self-dual linear code — The \([48,24,12]\) self-dual code is the only self-dual doubly even code at its parameters [1].
- Divisible code — The \([48,24,12]\) self-dual code is the only self-dual doubly even code at its parameters [1].

## Cousins

- Combinatorial design — Fixed-weight codewords of extremal self-dual doubly even codes whose length divides 24 form a combinatorial 5-design [3]. There are several designs associated with this code [4].
- Group-algebra code — The \([48,24,12]\) self-dual code is a group code for \(G\) being a dihedral group [5][6; Exam. 16.5.1].

## References

- [1]
- S. K. Houghten et al., “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
- [2]
- E. M. Rains and N. J. A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, eds. V. S. Pless and W. C. Huffman. Amsterdam: Elsevier, 1998, pp. 177–294.
- [3]
- E. F. Assmus Jr. and H. F. Mattson Jr., “New 5-designs”, Journal of Combinatorial Theory 6, 122 (1969) DOI
- [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

## Page edit log

- Victor V. Albert (2024-03-15) — most recent

## 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