Systematic code

In coding theory, a systematic code is one in which the input data are embedded in the encoded output. Similarly, a non-systematic code is one in which the output does not contain the input bits. Such codes are used to add redundant information to data; this allows errors to be detected (and possibly corrected, depending on the code used) when bits are lost or corrupted.

For example, transmitting data with a checksum is a type of systematic code: to encode the bit stream "01010110" with a simple two-bit checksum, one first calculates the checksum (01 + 01 + 01 + 10 = 01), then transmits the original stream followed by the new checksum bits: "0101011001" – note the existence of the original bits in the output.

Examples of systematic codes

• Data + checksum, the error detection used in IP.
• Reed-Solomon codes, the error correction used on CDs.

Examples of non-Systematic codes

• Fountain codes, popular for online content distribution.

Interesting theorem

Every linear block code is equivalent to a code that can be encoded be a systematic encoder

With this lemma we can decode a given codeword into its information word. Since our generator matrix G = [ I | P ], the first k symbols of our codeword are exactly our information word. If the generator matrix is not of the form [ I | P ], we transform it to generator matrix G', using column swaps. Performing the same column swaps on our code word, leads us to our information word.