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.
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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 with a systematic encoder[1]
With this lemma we can decode a given codeword into its information word. Since our generator matrix G = [ Ik | P ], the first k symbols of our codeword are exactly our information word (due to the identity matrix Ik). 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.
References
- ^ Richard E. Blahut (2003). Algebraic codes for data transmission (2nd ed.). Cambridge. Univ. Press. pp. 53–54. ISBN 9780521553742.
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