## Hamming Code Calculator

A **Hamming Code Calculator** is a tool used in digital communication and data storage to **detect** and **correct errors** in transmitted or stored data.

Named after its inventor

Richard Hamming, thiserror-correcting codeadds extra bits to the original data to create a new, longercode word. These additional bits, known asparity bits, allow the detection and correction ofsingle-bit errorsand the detection ofdouble-bit errors.

`We want to transmit the `**4-bit data word** 1011. A **Hamming Code Calculator** would add three **parity bits** to create the **7-bit code word** 1011010. If during transmission, one bit gets flipped (e.g., 1**0**11010 becomes 1**1**11010), the receiving end can **detect** and **correct** this error using the **Hamming code algorithm**.

## Hamming Code Chart

Data Bits | Parity Bits | Code Word |
---|---|---|

1011 | 010 | 1011010 |

0110 | 101 | 0110101 |

1101 | 011 | 1101011 |

0011 | 100 | 0011100 |

1111 | 000 | 1111000 |

## Hamming Code Formula

The **Hamming code formula** determines the number of **parity bits** (r) needed for a given number of **data bits** (m):

**2^r ≥ m + r + 1**

For example, if we have 4 data bits (m = 4), we solve:

2^r >= 4 + r + 1

2^3 = 8 >= 4 + 3 + 1 = 8

**Therefore, we need 3 parity bits for 4 data bits, resulting in a 7-bit code word.**

## What are Hamming codes?

**Hamming codes** are a family of **linear error-correcting codes** that can **detect** up to **two simultaneous bit errors** and **correct single-bit errors**. They are widely used in **computer memory systems**, such as **ECC RAM**, and in **digital communication** to ensure **data integrity**.

In a

(7,4) Hamming code,4 data bitsare encoded into a7-bit code word. If any single bit in the code word is corrupted during transmission or storage, theHamming codecan detect and correct this error, ensuring the original4-bit datais recovered accurately.

## How do Hamming codes work?

**Hamming codes** work by strategically placing **parity bits** within the data. These **parity bits** are calculated based on specific combinations of **data bits**. When decoding, the receiver recalculates the **parity bits** and compares them to the received values. Any discrepancy indicates an error, and the position of the error can be determined by analyzing which **parity bits** are incorrect.

For example, in a **(7,4) Hamming code**:

**Parity bit 1**covers bits 1, 3, 5, 7**Parity bit 2**covers bits 2, 3, 6, 7**Parity bit 4**covers bits 4, 5, 6, 7

**If an error occurs in bit 5, it will affect parity bits 1 and 4, allowing the receiver to pinpoint and correct the error.**

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