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--- tanszek:oktatas:techcomm:error_detection_and_correction [2024/10/07 05:13] – knehez
+++ tanszek:oktatas:techcomm:error_detection_and_correction [2024/11/12 07:33] (current) – [How Hamming Codes Work] knehez
@@ Line 13: / Line 13: @@
 |  0  |  0  |  1  |  **1**  |
 |  **1**  |  **1**  |  **1**  |  **1**  |
+Let's suppose that the first bit was wrong during transmission, then the parity bit in the row and column will show us the bad bit:
+|  1->0  |  0  |  1  |  **0** -> **1**  |
+|  0  |  1  |  1  |  **0**  |
+|  0  |  0  |  1  |  **1**  |
+|  **1** -> **0**  |  **1**  |  **1**  |  **1** -> **0**  |
@@ Line 24: / Line 32: @@
 $$ n = m + r $$
-If two code-words are given, for example : **0101110** and **001111110** and the only difference between them is 1 bit, then the 'Hamming distance' of these code-words will be 1. This is an interesting measure for distance because it does not matter which bit in the row is different or whether it belongs to a binary or decimal system. So for example : **45635263** and **45615263** have the Hamming distance of 1 too. It does not matter that 10 different digits could be in the place of the wrong digit.
+If two code-words are given, for example : **0101110** and **0111110** and the only difference between them is 1 bit, then the 'Hamming distance' of these code-words will be 1. This is an interesting measure for distance because it does not matter which bit in the row is different or whether it belongs to a binary or decimal system. So for example : **45635263** and **45615263** have the Hamming distance of 1 too. It does not matter that 10 different digits could be in the place of the wrong digit.
 The **Hamming-style** correction code supposed to increase the number of parity bits. To correct single bit errors we have to use **k** number of parity bits using this formula:
@@ Line 48: / Line 56: @@
 For example, in an 11-bit data block, we might have 4 parity bits, making it a 15-bit Hamming code.
-  * Detecting Errors: After transmission, the parity bits are recalculated and compared with the received parity bits. If there is any mismatch, the system can determine which bit is incorrect by checking the positions covered by the erroneous parity bits.
+  * **Detecting Errors**: After transmission, the parity bits are recalculated and compared with the received parity bits. If there is any mismatch, the system can determine which bit is incorrect by checking the positions covered by the erroneous parity bits.
-  * Correcting Errors: If a single-bit error is detected, the Hamming code identifies which bit is incorrect (from the parity bits' positions) and corrects it by flipping the bit.
+  * **Correcting Errors**: If a single-bit error is detected, the Hamming code identifies which bit is incorrect (from the parity bits' positions) and corrects it by flipping the bit.
 ==== Example ====