Description
A type of \(q\)-ary code whose parameters satisfy the Hamming bound with equality.
An \((n,K,2t+1)_q\) code is perfect if parameters \(n\), \(K\), \(t\), and \(q\) are such that the Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t}(q-1)^{j}{n \choose j}\leq q^{n}/K \tag*{(1)}\end{align} becomes an equality. In other words, the code's packing radius matches its covering radius.
For example, for a binary \(q=2\) 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 \(n\) \(q\)-ary strings.
Any perfect linear code is either a repetition code, a Hamming code, or a binary or ternary Golay code [1]. If \(q\) is a prime power, any distance-three code is either a Hamming code or a nonlinear code with the same parameters; see [2; pg. 100] for more details. There are many nonlinear perfect codes [3–15].
Cousins
- Combinatorial design— Perfect codes and combinatorial designs are related [16,17].
- Editing code— Perfect deletion correcting codes can be constructed using combinatorial design theory [18,19].
- \([6,3,4]_4\) Hexacode— The shortened hexacode is perfect [20; Exer. 578].
- Perfect quantum code— A classical (quantum) perfect code saturates the classical (quantum) Hamming bound.
Member of code lists
Primary Hierarchy
References
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- [15]
- 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
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- E. F. Assmus, Jr. and H. F. Mattson, Jr., “Coding and Combinatorics”, SIAM Review 16, 349 (1974) DOI
- [17]
- K. T. Phelps, “Combinatorial designs and perfect codes”, Electronic Notes in Discrete Mathematics 10, 220 (2001) DOI
- [18]
- P. A. H. Bours, “On the construction of perfect deletion-correcting codes using design theory”, Designs, Codes and Cryptography 6, 5 (1995) DOI
- [19]
- A. Mahmoodi, Designs, Codes and Cryptography 14, 81 (1998) DOI
- [20]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [21]
- J. Borges, J. Rifà, and V. A. Zinoviev, “On Completely Regular Codes”, (2017) arXiv:1703.08684
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 code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/perfect