Question

To provide more reliability than a single parity bit can give, an error- detecting coding scheme uses one parity bit for checking all the odd-numbered bits and a second parity bit for all the even-numbered bits. What is the Hamming distance of this code?

Short answer

The Hamming distance of this code is 2.

Step-by-step solution

Understanding Hamming Distance

The Hamming distance is the minimum number of single-bit changes required to convert one code word in a set into another.

Identifying the Parity Check Method

In this coding scheme, we have two parity bits: one for all the odd-numbered bits and another for all the even-numbered bits. This effectively creates a code word with two parity bits appended.

Analyzing Code Word Changes

Any single-bit error in the original data will be detected by one of the parity bits, changing at least the relevant parity bit. This means the code words are at least 2 apart for 1-bit errors.

Calculating the Hamming Distance

Since any single-bit error will be caught by the parity check (because it changes a parity bit), the code words differ by at least 2 bits in the error scenario. However, the code does not detect 2-bit errors, so the distance is 2.

Key concepts

Error-Detecting Code

When transmitting data, ensuring the integrity of that data is a crucial concern. An error-detecting code refers to a system designed to detect errors in the data during transmission. This does not correct the errors but flags them for review or retransmission. One common strategy for error detection involves using additional bits known as parity bits, which help verify the integrity of the data.
For instance, parity bits can be used to check specific sequences of bits, and if an error is detected, the data can be re-sent. This is effective for spotting single-bit errors but may require more complex schemes for detecting multiple errors. Multiple techniques like cyclic redundancy check (CRC) and checksums are also used to enhance the detection of errors along with parity bits.

Parity Bit

A parity bit is a simple error-detecting code mechanism, adding just a single bit to a set of binary data. This extra bit ensures that the total number of set bits (bits with a value of 1) is either even (even parity) or odd (odd parity).
For example, if we have a code word like 1101 and use an odd parity strategy, the parity bit will be 0, as 1101 already has an odd number of 1s. If we used an even parity strategy, the parity bit would instead be 1 to make the total even.

• **Even parity**: Total number of 1's in the data plus the parity bit is even.
• **Odd parity**: Total number of 1's in the data plus the parity bit is odd.

When data with a parity bit is transmitted, the receiving end verifies the integrity by checking the parity condition. If it doesn't match expectations, an error is detected.

Single-Bit Error

A single-bit error occurs when one bit within a data sequence gets flipped during transmission, changing a 0 to a 1 or vice versa. This type of error is the simplest and most common form of data corruption. Despite its simplicity, a single-bit error can have significant impacts depending on the data's nature and structure.
Error-detecting codes, like a parity bit scheme or others, excel at identifying single-bit errors. With a parity check, if a single bit is altered, the calculated parity will no longer match the expected parity, and thus, an error is detected immediately.

• **Detection**: Parity checks detect single-bit errors easily.
• **Implication**: Single-bit changes can critically alter data meaning or instructions.

Code Words

In the context of error detection and correction, code words refer to sequences of bits that represent data or information along with additional bits (such as parity bits) to encode and safeguard the data. In our example with parity bit codes, the code word consists of the original data plus the parity bits used for error detection.
The arrangement of these code words determines the ease with which errors can be detected or corrected. For example, using two parity checks—one for even-numbered bits and one for odd—significantly raises the probability of detecting single-bit errors. These issues underline why the Hamming distance, which denotes the minimum number of substitutions needed to change one code word into another, is vital.
Here, the code's design effectively allows us to identify mismatches swiftly, ensuring errors don't go unnoticed.