# Binary code

## Description

## Protection

A binary code \(C\) corrects \(t\) errors in the Hamming distance if \begin{align} \forall x \in C~,~D(x,x+y) < D(x' , x+y) \end{align} for all codewords \(x' \neq x\) and all \(y\) such that \(|y|=t\), where \(D\) is the Hamming distance and \(|y| = D(y,0) \). A code corrects \(t\) errors if and only if \(d \geq 2t+1\), i.e., a code corrects errors on \(t \leq \left\lfloor (d-1)/2 \right\rfloor\) coordinates. In addition, a code detects errors on up to \(d-1\) coordinates, and corrects erasure errors on up to \(d-1\) coordinates.

Performance of binary codes can also be measured with respect to deletions and insertions of bits into the codewords. In this case, the metric measuring distance of an error word to the nearest codeword is the deletion distance: given vectors \(u,v\), this distance is one-half the smallest number of deletions and insertions needed to change \(u\) to \(v\). A code \(C\) corrects \(e\) delections if all codewords are separated by at least \(e+1\) in the deletion distance [1].

## Decoding

## Parent

## Children

## Cousins

- Fock-state bosonic code — Fock-state code distance is a natural extension of Hamming distance between binary strings.
- Group-based code — Group-based codes whose alphabet is based on the group \(\mathbb{Z}_2\) are binary codes.
- Movassagh-Ouyang Hamiltonian code — Movassagh-Ouyang codes are constructed from classical binary codes.

## References

- [1]
- N. J. A. Sloane, “On Single-Deletion-Correcting Codes”. math/0207197

## Page edit log

- Victor V. Albert (2022-07-06) — most recent
- Victor V. Albert (2022-02-16)
- Victor V. Albert (2021-10-29)

## Zoo code information

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/bits/bits_into_bits.yml.