Abstract
SLEPIAN 1 has pointed out that all systematic binary codes of length n form a sub-group of the group of all binary words of length n. The problem of choosing a systematic code for correcting a given set of errors is thus equivalent to finding a sub-group such that each error to be corrected for lies in separate cosets. Fire2 has pointed out that the sum modulo 2 of two parity check sequences corresponding to two error patterns is the same as the parity check sequence corresponding to the sum modulo 2 of the same two error patterns. Also, each parity check sequence is uniquely associated with one coset. This sets up an isomorphism between the group of parity check sequences and the quotient group of the code group in the group of all words.
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References
Slepian, D., Bell Syst. Tech. J., 35, 203 (1956).
Fire, P., Technical Report No. 55, Stanford Electronics Laboratories, Stanford University (April 24, 1959).
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BANERJI, R. A Systematic Method for the Construction of Error-correcting Group Codes. Nature 186, 627 (1960). https://doi.org/10.1038/186627a0
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DOI: https://doi.org/10.1038/186627a0
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