## Description

A type of binary code whose parameters satisfy the Johnson bound with equality.

An \((n,K,2t+1)\) binary code is nearly perfect if parameters \(n\), \(K\), and \(t\) are such that the Johnson bound [1], \begin{align} \frac{{n \choose t}\left(\frac{n-t}{t+1}-\left\lfloor \frac{n-t}{t+1}\right\rfloor \right)}{\left\lfloor \frac{n}{t+1}\right\rfloor }+\sum_{j=0}^{t}{n \choose j}\leq2^{n}/K \tag*{(1)}\end{align} becomes an equality ([4], Sec. 2.3.5; see also Ref. [5], Ch. 17). All nearly perfect binary codes are either perfect, or correspond to either Preparata codes or one of the \((2^{2^r-2-r},2^r-2,3)\) codes for \(r\geq 3\) [6].

Similar definitions can be made for \(q\)-ary codes, but all nearly perfect \(q\)-ary codes must be perfect [7,8].

## Parents

- Binary code
- Quasi-perfect code — Nearly perfect codes are quasi-perfect [5; pg. 533].

## Children

- Perfect binary 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.
- Repetition code
- Preparata code — Preparata codes are uniformly packed and nearly perfect [2]. For any word \(u\) and Preparata codebook \(C\) with \(d(u, C) > 2\), we have that \(u\) has a distance 2 or 3 to exactly \(\left\lfloor (2^{m+1}-1)/3\right\rfloor\) codewords.
- Single parity-check (SPC) code

## Cousins

- Golay code — The extended Golay code is nearly perfect.
- \([2^r-1,2^r-r-1,3]\) Hamming code — Shortened Hamming codes \([2^r-2,2^r-r-2,3]\) are nearly perfect ([5], pg. 533).

## References

- [1]
- S. Johnson, “A new upper bound for error-correcting codes”, IEEE Transactions on Information Theory 8, 203 (1962) DOI
- [2]
- J. M. Goethals and S. L. Snover, “Nearly perfect binary codes”, Discrete Mathematics 3, 65 (1972) DOI
- [3]
- N. V. Semakov, V. A. Zinov'ev, G. V. Zaitsev, “Uniformly Packed Codes”, Probl. Peredachi Inf., 7:1 (1971), 38–50; Problems Inform. Transmission, 7:1 (1971), 30–39
- [4]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [5]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [6]
- Kauko Lindström. "The nonexistence of unknown nearly perfect binary codes." PhD diss., Turun yliopisto, 1975.
- [7]
- K. Lindstrom and M. J. Aaltonen, "The nonexistence of nearly perfect nonbinary codes for 1 =< e =< 10", Ann. Univ. Turku, Ser. A I, No. 172, 1976.
- [8]
- K. Lindström, “All nearly perfect codes are known”, Information and Control 35, 40 (1977) DOI

## Page edit log

- Victor V. Albert (2022-07-19) — most recent

## Cite as:

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