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--- tanszek:oktatas:techcomm:reed-solomon_codes [2024/10/06 19:37] – [Step 8: Error Detection and Correction] knehez
+++ tanszek:oktatas:techcomm:reed-solomon_codes [2024/10/06 19:43] (current) – [Step 1: Data Symbols] knehez
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 ~~NOTOC~~
 ==== Reed-Solomon codes ====
+**Reed-Solomon (RS) codes** are a type of **error-correcting code** used to detect and correct data transmission or storage errors. They are particularly effective at correcting burst errors, where multiple consecutive bits are corrupted. Reed-Solomon codes are widely used in applications like **CDs, DVDs, QR codes, satellite communications**, and **data storage systems**.
+==== Key Concepts ====
+=== 1. Symbols Instead of Bits ===
+Unlike simpler error-correcting codes (such as Hamming codes, which operate on individual bits), Reed-Solomon codes operate on **symbols**. A symbol is a block of bits (e.g., 8 bits) that represents a larger unit of data.
+For example, in an 8-bit RS code, each symbol is an 8-bit value, allowing the code to work on groups of 8 bits (1 byte) at a time. RS codes can handle burst errors, meaning they can correct errors that affect multiple consecutive bits in a single symbol.
+=== 2. Block-Based Code ===
+Reed-Solomon codes are **block codes**, meaning they work by dividing data into fixed-size blocks and encoding each block independently.
+  * **Message Length**: The number of data symbols in a block.
+  * **Codeword Length**: The total number of symbols (including parity) in a block after encoding.
+For example, in a typical RS code, you might have a message length of `k` data symbols and a codeword length of `n` total symbols, where `n > k`. The difference (`n - k`) represents the number of **redundant (parity)** symbols added for error correction.
+=== 3. Error Correction Capability ===
+Reed-Solomon codes can detect and correct multiple symbol errors. Specifically, an RS code can correct up to `(n - k) / 2` symbols of errors in a block.
+  * Error correction capability: The number of correctable symbols depends on how many parity symbols are added. More parity symbols increase the error-correcting strength but also increase the overhead.
 We'll use a simplified version with small numbers to make the process easier to understand. Typically, Reed-Solomon codes use larger fields (like 255 symbols for CDs), but in this example, we'll use a smaller field size.
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 Let’s assume we want to encode the following **3 data symbols**:
 \[
-\text{Data} = \{ 6, 4, 3 \}
+\text{data} = \{ 6, 4, 3 \}
 \]