Question

Sixteen-bit messages are transmitted using a Hamming code. How many check bits are needed to ensure that the receiver can detect and correct single-bit errors? Show the bit pattern transmitted for the message 1101001100110101 . Assume that even parity is used in the Hamming code.

Short answer

Their message requires 5 check bits. The transmitted pattern is 001010101100010011011.

Step-by-step solution

Understanding Hamming Code

Hamming code is a method used to detect and correct single-bit errors in transmitted data. To do this, it introduces check bits into the data stream. The formula for determining the number of check bits (\(r\)) is given by the inequality \(2^r \geq m + r + 1\), where \(m\) is the number of data bits and \(r\) is the number of check bits.

Calculate the Number of Check Bits

For the given 16-bit message, we need to find \(r\) such that \(2^r \geq 16 + r + 1\). Testing various values of \(r\):- \(r = 4\): \(2^4 = 16\), \(16 ot\geq 17\)- \(r = 5\): \(2^5 = 32\), \(32 \geq 21\)Thus, 5 check bits are needed.

Calculate Check Bit Positions

The positions of the check bits in the transmission are powers of 2: 1, 2, 4, 8, and 16. These positions are reserved in the final bit sequence to contain the parity bits.

Determine Check Bit Values

Insert the check bits into their positions: _ _ 1 _ 1 0 1 0 _ 1 1 0 0 _ 1 0 1 0 1. Calculate each check bit (C1, C2, C4, C8, C16) for even parity: - C1 checks positions (1, 3, 5, 7, 9, 11, 13, 15): 1, 1, 1, 0, 0, 0, 1 = 4 ones, C1 = 0 - C2 checks positions (2, 3, 6, 7, 10, 11, 14, 15): 0, 1, 0, 1, 1, 0, 1 = 4 ones, C2 = 0 - C4 checks positions (4, 5, 6, 7, 12, 13, 14, 15): 1, 0, 0, 1, 0, 1, 1 = 4 ones, C4 = 0 - C8 checks positions (8, 9, 10, 11, 12, 13, 14, 15): 0, 1, 1, 0, 0, 1, 1 = 4 ones, C8 = 0 - C16 checks positions (16): 1 is a data bit. Thus: C1 = 0, C2 = 0, C4 = 0, C8 = 0, and C16 = 0.

Construct the Final Bit Pattern

Place the calculated check bits into their positions to complete the 21-bit sequence: 00 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1. This is the bit pattern transmitted for the message, ensuring single-bit error detection and correction.

Key concepts

Error Detection

Error detection is a crucial process in data communication. It ensures that the information being transferred from one point to another is accurate and free from any unintended alterations. In simple terms, error detection identifies whether errors have occurred during the transmission process.

For instance, when using a Hamming code, a method specifically developed for communication systems, we can detect single-bit errors with precision. Hamming codes leverage extra bits known as check bits or parity bits to identify errors. By embedding these bits at strategic positions within a data message, the receiving system can determine if any bits have been switched (from 0 to 1 or vice versa) due to transmission noise or interference.

Whenever an error is detected, typically indicated by mismatches in expected parity conditions, the system can take action to inform the sender to resend the data, or proceed with error correction techniques if supported.

Error Correction

While error detection is about recognizing that a problem exists, error correction is about fixing that problem. Hamming codes not only detect that an error has occurred but also identify the exact position of the single-bit error.

Hamming codes achieve error correction by using a combination of check bits and data structure. Each check bit covers a specific set of data bits, and when the message arrives, the receiver utilizes these check bits to pinpoint errors. The math involved here allows the system to recognize which exact bit is incorrect, thus enabling the immediate switching of the incorrect bit to its opposite value (0 to 1 or 1 to 0).

This ability to correct errors automatically is invaluable, especially in digital communications, where the integrity of data is key. For example, in the case of a 16-bit message formatted with Hamming code, detailed calculations are used to assure that any single-bit mistake can be instantly rectified without delay. This ensures data reliability and reduces the need for re-transmissions.

Parity Bits

Parity bits are the backbone of error detection and correction, especially when using methods like Hamming code. Parity bits add an extra layer of information to your data to verify its integrity.

To simplify, parity bits are calculated as checksums for subsets of data bits. In even parity, which is often used with Hamming codes, these bits are set so that the total number of 1s in the bit subset, including the parity bit, is even. If a single-bit error occurs, the parity condition will be violated, thereby signaling the error's presence.

In the context of a 16-bit message, adding sufficient parity bits allows us to ensure that each part of the message adheres to the predefined parity rule. This not only allows for the detection of an error but also provides the ability to locate it within the message based on which parity bits are compromised.

Check Bits

Check bits, often interchangeably referred to as parity bits in the context of Hamming codes, play a crucial role in error control during data transmission. These bits are strategically positioned throughout the data message to check for errors.

Check bits are placed at specific locations that are powers of 2 in the sequence of data, such as 1, 2, 4, 8, and so on. Each check bit is responsible for checking a certain combination of bits, and together, they create a system that ensures every bit of data is verified.

When you calculate these check bits, such as in a 21-bit sequence of a 16-bit message plus 5 check bits in the Hamming code, you ensure that the format allows for both error detection and correction. This strategic setup ensures that each piece of data can self-verify its integrity, promoting reliable communication.