## Description

An \((n,K,2t+1)\) binary code is perfect if parameters \(n\), \(K\), and \(t\) are such that the binary Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t} {n \choose j} \leq 2^{n}/K \tag*{(1)}\end{align} becomes an equality. For example, for a code with one logical bit (\(K=2\)) and \(t=1\), the bound becomes \(n+1 \leq 2^{n-1}\). Perfect codes are those for which balls of Hamming radius \(t\) exactly fill the space of all \(n\) binary strings.

Any perfect linear binary code is either a binary repetition code, a binary Hamming code, or the binary Golay code [1].

For codes with \(K=2^k\), one can work out an asymptotic Hamming bound in the large-\(n,k,t\) limit, \begin{align} \frac{k}{n}\leq 1-h(t/n), \tag*{(2)}\end{align} where \(h\) is the binary entropy function.

There are many inequivalent nonlinear perfect binary codes [2–6]; for example, there are 5983 inequivalent perfect distance-three codes of length 15 [7]. Nonlinear perfect codes can have arbitrary finite groups as their automorphism groups [8], including the trivial group [9,10]. The automorphism group of a distance-three perfect binary code is no greater than the automorphism of the Hamming code of the same length [11–14].

## Parents

- Nearly perfect code — Perfect binary codes are nearly perfect, and \(t+1\) divides \(n-t\) for such codes. In addition, any perfect code can be extended to a nearly perfect code.
- Perfect code
- Orthogonal array (OA) — Perfect distance-three binary codes of length \(n =2^m-1\) are equivalent to binary orthogonal arrays of strength \(t = 2^{m-1}-1\) [5,15,16].

## Children

- Golay code — The Golay code is perfect.
- \([2^r-1,2^r-r-1,3]\) Hamming code
- Vasilyev code — Vasilyev codes are perfect nonlinear binary codes and are inequivalent to any linear code.

## Cousins

- Repetition code — Repetition codes are perfect for odd \(n\).
- Cycle code — A family of cycle codes saturate the asymptotic Hamming bound [17].

## References

- [1]
- K. Lindström, “All nearly perfect codes are known”, Information and Control 35, 40 (1977) DOI
- [2]
- J. L. Vasilyev On nongroup close-packed codes (in Russian), Probl. Kibernet., 8 (1962), 337-339, translated in Probleme der Kibernetik 8 (1965), 375-378.
- [3]
- D. S. Krotov, “Lower bounds for the number of m-quasigroups of order four and of the number of perfect binary codes”, Diskretn. Anal. Issled. Oper., Ser. 1, 7:2 (2000), 47–53
- [4]
- K. T. Phelps, “A Combinatorial Construction of Perfect Codes”, SIAM Journal on Algebraic Discrete Methods 4, 398 (1983) DOI
- [5]
- T. Etzion and A. Vardy, “Perfect binary codes: constructions, properties, and enumeration”, IEEE Transactions on Information Theory 40, 754 (1994) DOI
- [6]
- D. S. Krotov and S. V. Avgustinovich, “On the Number of \(1\)-Perfect Binary Codes: A Lower Bound”, IEEE Transactions on Information Theory 54, 1760 (2008) arXiv:math/0608278 DOI
- [7]
- P. Ostergard and O. Pottonen, “The Perfect Binary One-Error-Correcting Codes of Length <formula formulatype="inline"><tex Notation="TeX">\(15\)</tex></formula>: Part I—Classification”, IEEE Transactions on Information Theory 55, 4657 (2009) arXiv:0806.2513 DOI
- [8]
- K. T. Phelps, “Every finite group is the automorphism group of some perfect code”, Journal of Combinatorial Theory, Series A 43, 45 (1986) DOI
- [9]
- S. V. Avgustinovich and F. I. Solov’eva, “Perfect binary codes with trivial automorphism group”, 1998 Information Theory Workshop (Cat. No.98EX131) DOI
- [10]
- O. Heden et al., “On the symmetry group of extended perfect binary codes of length \(n+1\) and rank \(n-\log(n+1)+2\)”, Advances in Mathematics of Communications 6, 121 (2012) DOI
- [11]
- F. I. Solov'eva, S. T. Topalova, “On Automorphism Groups of Perfect Binary Codes and Steiner Triple Systems”, Probl. Peredachi Inf., 36:4 (2000), 53–58; Problems Inform. Transmission, 36:4 (2000), 331–335
- [12]
- F. I. Solov'eva, S. T. Topalova, “Perfect binary codes and Steiner triple systems with maximum orders of automorphism groups”, Diskretn. Anal. Issled. Oper., Ser. 1, 7:4 (2000), 101–110
- [13]
- S. A. Malyugin, “On the order of the automorphism group of perfect binary codes”, Diskretn. Anal. Issled. Oper., Ser. 1, 7:4 (2000), 91–100
- [14]
- O. Heden, “On the size of the symmetry group of a perfect code”, Discrete Mathematics 311, 1879 (2011) DOI
- [15]
- P. Delsarte, “Four fundamental parameters of a code and their combinatorial significance”, Information and Control 23, 407 (1973) DOI
- [16]
- P. R. J. Ostergard, O. Pottonen, and K. T. Phelps, “The Perfect Binary One-Error-Correcting Codes of Length 15: Part II—Properties”, IEEE Transactions on Information Theory 56, 2571 (2010) arXiv:0903.2749 DOI
- [17]
- H. Bombin and M. A. Martin-Delgado, “Homological error correction: Classical and quantum codes”, Journal of Mathematical Physics 48, (2007) arXiv:quant-ph/0605094 DOI

## Page edit log

- Victor V. Albert (2022-07-19) — most recent
- Mustafa Doger (2022-04-01)
- Victor V. Albert (2022-03-21)
- Victor V. Albert (2021-12-01)

## Cite as:

“Perfect binary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/perfect_binary