For the binary code C = {01101 , 00011 , 10110 , 11000}, use the nearest neighbour decoding rule to decode the following received words:
00000, (b) 01111, (c) 10110, (d) 10011, (e) 11011.
The Correct Answer and Explanation is :
The nearest neighbor decoding rule is a method used in coding theory to correct errors in received messages. It operates on the principle of selecting the codeword from a predefined set that is closest to the received word, typically measured by the Hamming distance—the number of positions at which the corresponding symbols are different.
Given Codewords:
The binary code ( C ) consists of the following codewords:
- 01101
- 00011
- 10110
- 11000
Received Words and Decoding:
- Received Word: 00000
- Hamming Distances:
- 00000 vs. 01101: 3 errors
- 00000 vs. 00011: 2 errors
- 00000 vs. 10110: 4 errors
- 00000 vs. 11000: 2 errors
- Decoded Codeword: 00011 or 11000 (both have the same Hamming distance of 2)
- Received Word: 01111
- Hamming Distances:
- 01111 vs. 01101: 1 error
- 01111 vs. 00011: 3 errors
- 01111 vs. 10110: 3 errors
- 01111 vs. 11000: 3 errors
- Decoded Codeword: 01101
- Received Word: 10110
- Hamming Distances:
- 10110 vs. 01101: 3 errors
- 10110 vs. 00011: 4 errors
- 10110 vs. 10110: 0 errors
- 10110 vs. 11000: 3 errors
- Decoded Codeword: 10110
- Received Word: 10011
- Hamming Distances:
- 10011 vs. 01101: 3 errors
- 10011 vs. 00011: 1 error
- 10011 vs. 10110: 2 errors
- 10011 vs. 11000: 3 errors
- Decoded Codeword: 00011
- Received Word: 11011
- Hamming Distances:
- 11011 vs. 01101: 3 errors
- 11011 vs. 00011: 2 errors
- 11011 vs. 10110: 3 errors
- 11011 vs. 11000: 1 error
- Decoded Codeword: 11000
Explanation:
In nearest neighbor decoding, the objective is to identify the codeword that is closest to the received word in terms of Hamming distance. The Hamming distance between two binary strings is the number of positions at which the corresponding symbols differ.
For each received word, we compute the Hamming distance to each codeword in the set ( C ). The codeword with the smallest Hamming distance is selected as the decoded codeword.
In cases where multiple codewords have the same minimum Hamming distance to the received word, the decoding rule may not uniquely determine the correct codeword. This ambiguity can occur due to the structure of the code and the nature of the errors introduced during transmission.
For example, when the received word is 00000, both 00011 and 11000 have the same Hamming distance of 2. In such situations, additional information or error-correcting techniques may be necessary to resolve the ambiguity and accurately decode the received word.
The effectiveness of nearest neighbor decoding depends on the minimum Hamming distance between codewords in the code. A larger minimum distance allows the code to correct more errors but may reduce the code rate, which is the ratio of the number of information bits to the total number of bits in the codeword. Therefore, there is a trade-off between error correction capability and code rate that must be considered when designing error-correcting codes.
In summary, nearest neighbor decoding is a fundamental technique in coding theory that utilizes Hamming distance to correct errors in received messages. While it is effective in many scenarios, its performance can be influenced by the code’s structure and the nature of the errors encountered during transmission.