Binary repetition code[1]


\([n,1,n]\) binary linear code encoding one bit of information into an \(n\)-bit string. The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information. The idea is to increase the code distance by repeating the logical information several times. It is a \((n,1)\)-Hamming code.


Detects errors on up to \(\frac{n-1}{2}\) coordinates, corrects erasure errors on up to \(\frac{n-1}{2}\) coordinates. The generator matrix is \(G=\left[\begin{smallmatrix}1 & 1&\cdots& 1 & 1 \end{smallmatrix}\right]\).


Code rate is \(\frac{1}{n}\), code distance is \(n\).


Calculate the Hamming weight \(d_H\) of the code. If \(d_H\leq \frac{n-1}{2}\), decode the code as 0. If \(d_H\geq \frac{n+1}{2}\), decode the code as 1.


Suppose each bit has probability \(p\) of being received correctly, independent for each bit. The probability that a repetition code is received correctly is \(\sum_{k=0}^{(n-1)/2}\frac{n!}{k!(n-k)!}p^{n-k}(1-p)^{k}\). If \(\frac{1}{2}\leq p\), then people can always increase the probability of success by increasing the number of physical bit \(n\).


Repetition codesm in conjunction with other codes, were used in magnetic disks [2].



Zoo code information

Internal code ID: repetition

Your contribution is welcome!

on (edit & pull request)

edit on this site

Zoo Code ID: repetition

Cite as:
“Binary repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_repetition, title={Binary repetition code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Permanent link:


W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
T. Klove and M. Miller, “The Detection of Errors After Error-Correction Decoding”, IEEE Transactions on Communications 32, 511 (1984). DOI

Cite as:

“Binary repetition code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.