Assignment Topic: Hamming Code for Error Detection and Correction

Objective: To understand and apply the principles of Hamming Code for detecting and correcting single$-$bit errors in digital data.

Task:

$1.$ Explain the basics of the Hamming Code, including the following details:

Construction of Hamming Code:

$-$ How to calculate the number of parity bits $($r$)$ required

$-$ How to determine the position of parity bits $($powers of $2)$

$-$ How to construct the Hamming Code by combining data bits and parity bits

Parity Bit Calculation:

$-$ How to calculate each parity bit using XOR operation

$-$ How to use the data bits to calculate the parity bits

Error Detection and Correction:

$-$ How to detect a single$-$bit error using the parity bits

$-$ How to identify the location of the error

$-$ How to correct the error by flipping the erroneous bit

Hamming Code Properties:

$-$ The minimum distance between codewords $(3)$

$-$ The ability to detect and correct single$-$bit errors

$-$ The inability to detect or correct multiple$-$bit errors:

$-$ The advantages and limitations

$2.$ Calculate the Hamming Code for a given data set $($provided below$)$:

$-$ Data bits: $1011$

$-$ Parity bits: $???($calculate and fill in the correct values$)$

$3.$ Assume a single$-$bit error occurs during transmission, and the received data is:

$-1001$ XXX $($X represents the parity bits$)$

$-$ Detect the error using the Hamming Code

$-$ Correct the error and provide the original data bits

Requirements:

$-$ Answer all questions and tasks clearly and concisely

$-$ Use proper grammar, spelling, and punctuation

$-$ Show all calculations and work for the Hamming Code

$-$ Use diagrams or examples to illustrate your points, if necessary