The Simplest Form of Protection

Repetition codes are the most intuitive and fundamental form of Forward Error Correction (FEC). The core idea is simple and mirrors how we communicate in everyday life: if you want to make sure someone hears you correctly in a noisy room, you repeat yourself. Repetition codes do exactly that for digital data.

The general principle is to transmit each individual bit of the original message a predetermined number of times. This introduces redundancy, which can be used by the receiver to overcome errors introduced by the channel.

How It Works: The R3 Repetition Code

The most common example is the R3 code, also known as the (3,1) repetition code. Here, every single bit of information is repeated three times.

• To send a logical '0', the transmitter sends the sequence 000.
• To send a logical '1', the transmitter sends the sequence 111.

Decoding by Majority Vote

At the receiving end, the decoder examines each three-bit block and makes a decision based on a simple "majority vote":

Error Correction Capabilities and Limitations

The power of repetition codes lies in their ability to correct errors. If a single bit in a three-bit block gets flipped by noise during transmission, the majority vote will still yield the correct original bit.

Example of successful correction:
1. Sender wants to send: 1
2. Encoded and transmitted signal: 111
3. Noise corrupts the signal. Received signal: 101
4. Decoder's majority vote (two 1s, one 0) → Decodes correctly to: 1. The error was corrected!

Example of failure:
However, if two bits are flipped, the majority vote will lead to an incorrect decision.
1. Transmitted signal: 111
2. Heavy noise corrupts the signal. Received signal: 001
3. Decoder's majority vote (one 1, two 0s) → Decodes incorrectly to: 0.

Formal Capabilities (Hamming Distance)

The error-handling capabilities of a code are determined by its minimum Hamming distance ($d_{min}$). For the R3 code, the only valid codewords are $000$ and $111$, and they differ in 3 positions, so $d_{min}=3$.

• The number of errors a code can detect is $d = d_{min} - 1$. For R3, it's $3 - 1 = 2$ errors.
• The number of errors a code can correct is $t = \lfloor \frac{d_{min} - 1}{2} \rfloor$. For R3, it's $\lfloor \frac{3 - 1}{2} \rfloor = 1$ error.

Interactive Repetition Code (R3/R5/R7)

Advantages and (Significant) Disadvantages

Advantages

• Simplicity: They are incredibly easy to understand, implement in hardware, and analyze.

Disadvantages

• Extreme Inefficiency: This is their greatest weakness. The code rate, which is the ratio of information bits to total transmitted bits, is very low. For the R3 code, the rate is $\frac{1}{3}$. This means that two-thirds of the transmission is redundant overhead. Sending a 1 MB file requires transmitting 3 MB of data.
• Low Error Correction Capability: They can only handle a small number of random errors. They are very vulnerable to burst errors, where multiple consecutive bits are corrupted.
• Increased Delay and Bandwidth: Transmitting three times as many bits naturally triples the time needed or requires three times the bandwidth for the same effective data rate, making them unsuitable for high-speed communication.