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--- tanszek:oktatas:techcomm:error_detection_and_correction [2024/10/06 18:45] – [Example] knehez
+++ tanszek:oktatas:techcomm:error_detection_and_correction [2024/11/12 07:33] (current) – [How Hamming Codes Work] knehez
@@ Line 1: / Line 1: @@
+===== Elias-style block protection =====
+Elias-style block protection uses horizontal and vertical control bits. It should be used if the protected data can be written in matrix form. During the decoding process, the logical values of the equations are examined both individually and combined.
+**Example**
+A binary data which is stored in a 3x3 matrix is given: **101011001**
+Let's write it down in matrix form and attach parity bits, too.
+|  1  |  0  |  1  |  **0**  |
+|  0  |  1  |  1  |  **0**  |
+|  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**  |
 ===== Error Detection and Correction Using Hamming Codes =====
@@ Line 7: / 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 31: / 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 ====
@@ Line 82: / Line 107: @@
 The common values of the first and second parity bits are the following: 1, 2, 3, 5, 6, 7. The defective one has to be among these.
-However from 5, 6, 7 are included in the good parity bits -> therefore the wrong parity bit is the **3rd**.  (1, 2 cannot be wrong in this case. If 1 is wrong, only the p1 parity bit would be wrong.)
+However from 5, 6, 7 are included in the good parity bits -> therefore the wrong parity bit is the **3rd** one.  (note that: 1, 2 cannot be wrong in this case. If 1 was wrong, only the **p1** parity bit would be wrong.)