Reed–Solomon Interactive Simulator

GF(2³) Engine | RS(7,3) Code | 2-Symbol Error Correction

1. Source Bitstream (Input 9 bits)

Each 3 bits form 1 symbol in GF(2³). Total 3 data symbols.

2. Wireless Channel (Flip bits to test correction)

Click any bit below to simulate noise in the air.

STATUS: Clean Signal Errors: 0 symbols

3. Mathematical Decoder

Syndromes ($S_1 \dots S_4$):
0, 0, 0, 0

Detected Error Positions:
None

4. Recovered Data

101011000

✓ Matches Source

Decoder Execution Log

Step-by-Step Walkthrough: Input 101011000

1. Bit-to-Symbol Conversion Reed-Solomon works on Symbols, not bits. In GF(2³), each symbol is 3 bits.
Input: 101 | 011 | 000
Decimal Values: 5 3 0

2. Encoding (Adding Redundancy) To protect these 3 symbols, the RS(7,3) encoder adds 4 parity symbols.
Data: [5, 3, 0]
Calculated Parity: [6, 4, 2, 5]
Full Codeword: 530 + 6425

3. The Channel (Simulating Interference) Imagine the first bit flips from 1 to 0 during transmission:
Sent Bitstream: 101...
Received Bitstream: 001...
The first symbol is now 1 instead of 5.

4. The Decoding Rescue The decoder receives: [1, 3, 0, 6, 4, 2, 5]. It performs three math checks:

• Syndromes: It calculates a "check value." Since it isn't zero, it knows an error exists.
• Location: The Berlekamp-Massey algorithm identifies that the error is at Index 0.
• Correction: The Forney algorithm determines the error magnitude is 4. (1 XOR 4 = 5).

5. Final Recovery The error is fixed, the parity is discarded, and the symbols are turned back into bits.
Corrected Symbols: [5, 3, 0]
Output Bitstream: 101011000

Why 2 Errors? In RS(n, k), the number of errors you can fix is t = (n - k) / 2.
Here: (7 symbols - 3 data) / 2 = 2 symbols. You can flip multiple bits inside a single symbol, and it still only counts as "1 error."

Under the Hood: How the Decoder Works

The GPS Analogy: Why 4 syndromes for 1 error? Think of Syndromes like GPS satellites. To find exactly where a car is on a map, you need 3 or 4 satellites. One syndrome tells you there is an error; four syndromes tell you exactly where and how big the errors are.

Step 1: Syndrome Calculation (The Detectors)

The decoder plugs the received data into the Galois Field. If the result isn't zero, an error exists.

Received Syndromes: 6 1 5 4

Even for one error, all 4 syndromes "vibrate" because they are all mathematically linked to every position in the codeword.

Step 2: Location Search (The Scanner)

The decoder creates an Error Locator Polynomial. For your example, it finds the "pattern" in the syndromes and creates this equation:

Λ(x) = 1 + 2x

It then tests every position (Index 0 to 6). It finds that when x = Index 1, the equation equals zero. Position identified!

Step 3: Calculating Magnitude (The Repair)

Now we know the error is at Index 1. We use the Forney Algorithm to find out how to fix it. It calculates the "Error Magnitude" by comparing the syndromes against the identified position.

Magnitude = S₁ / (Position Factor)
Magnitude = 6 / 2 (in GF math) = 3

Step 4: The Final Correction

The decoder performs a final XOR (bit-wise addition) between what was received and the calculated error magnitude.

Received Symbol (1) ⊕ Magnitude (3) = Original Symbol (5)

Data is now restored to its original state: 101 (Binary for 5).

Why do we need 4? Reed-Solomon RS(7,3) is designed to fix up to t = 2 symbol errors. To find 2 errors, you need 2 equations for locations and 2 equations for magnitudes. That is why 4 syndromes are required!

Return to Online Simulations Main Page →

Information and Coding Theory Main Page →