Terminologies used in Error Detection and Error Correction

Error Detection vs Error Correction

• Error Detection: Receiver identifies that an error occurred but may not correct it.
  Example: Single parity bit.
• Error Correction: Receiver identifies and fixes the error using redundancy.
  Example: Hamming code, Reed-Solomon code.

Codeword

A codeword is the sequence of bits transmitted over a channel, consisting of data bits and redundancy bits. These redundancy bits allow the receiver to detect and correct errors.

Example: If your data is 1011 and you add 3 parity bits 011, the transmitted codeword becomes 1011011.

Data Bits vs Parity Bits

• Codeword length (n): Total bits transmitted = Data bits + Parity bits. Usually we define the codeword length first, then derive the parity bits (as in the example).
• Data bits (k): Original information to transmit.
• Parity bits (n-k): Extra bits added for error detection/correction.

Example: A Hamming (7,4) code has 4 data bits and 3 parity bits → total codeword length = 7.

Hamming Distance

The Hamming distance is the number of bit positions in which two codewords differ.

• With a hamming distance of d_min, you can do the following,
  • $t = \left\lfloor \frac{d_{\min} - 1}{2} \right\rfloor$ → can correct t errors
  • $t = d_{\min} - 1$ → can detect t errors

Example: 1011 vs 1111 differ by 1 bit → Hamming distance = 1.

Bit Error Rate (BER)

BER is the fraction of bits received incorrectly over total transmitted bits. FEC reduces effective BER by correcting errors before data reaches higher layers.

Redundancy / Coding Rate

• Redundancy: Extra bits added to help correct errors.
• Coding Rate (R): Ratio of data bits to total bits: R = k / n
  • High rate → fewer parity bits, less error protection
  • Low rate → more parity bits, stronger protection

Soft Decision vs Hard Decision

• Hard Decision: Receiver decides each bit as 0 or 1, then applies error correction.
• Soft Decision: Receiver uses probability/confidence of each bit → better error correction performance.

Burst Errors vs Random Errors

• Random Error: Single-bit errors scattered across codewords.
• Burst Error: Consecutive multiple-bit errors.
• Some codes (like Reed-Solomon) are designed to correct burst errors efficiently.

Generator Matrix & Parity-Check Matrix

• Generator Matrix (G): Encodes data bits into codewords.
• Parity-Check Matrix (H): Used to detect errors at the receiver.

Example: In Hamming (7,4) code, G converts 4 data bits into a 7-bit codeword; H is used to check parity and find errors.