Linear Block Codes: Encoding and Syndrome Decoding

The previous chapter defined some properties of linear block codes and discussed two examples of linear block codes (rectangular parity and the Hamming code), but the approaches presented for decoding them were specific to those codes. Here, we will describe a general strategy for encoding and decoding linear block codes. The decoding procedure we describe is syndrome decoding, which uses the syndrome bits introduced in the previous chapter. We will show how to perform syndrome decoding efficiently for any linear block code, highlighting the primary reason why linear (block) codes are attractive: the ability to decode them efficiently. We also discuss how to use a linear block code that works over relatively small block sizes to protect a packet (or message) made up of a much larger number of bits. Finally, we discuss how to cope with burst error patterns, which are different from the BSC model assumed thus far. A packet protected with one or more coded blocks needs a way for the receiver to detect errors after the error correction steps have done their job, because all errors may not be corrected. This task is done by an error detection code, which is generally distinct from the correction code. For completeness, we describe the cyclic redundancy check (CRC), a popular method for error detection.