# Single parity-check code

## Description

An \([n,n-1,2]\) binary linear error-detecting code encoding an \(n\)-bit codeword into an \((n+1)\)-bit string. In this code, parity information of a codeword is sotred in an extra parity bit. If the Hamming weight of a codeword is odd, then its parity is 1. If the Hamming weight of a codeword is even, then its parity is 0. This code is inexpensive since it only requires an extra parity bit and a single parity check.

## Protection

This code cannot protect information, it can only detect 1-bit error.

## Rate

The code rate is \(\frac{n}{n+1}\to 1\) as \(n\to\infty\). The code distance is 2.

## Encoding

Concatenate the codeword with a parity bit which encodes the parity information of codeword.

## Decoding

If the receiver finds that the parity information of a codeword disagrees with the parity bit, then the receiver will discard the information and request a resend.

## Realizations

Can be realized on almost every communication device.

## Parent

## Cousin

- Binary repetition code — Repetition code is dual to the single-parity check code.

## Zoo code information

## Cite as:

“Single parity-check code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/single_parity_check