A linear circuit viewpoint on error-correcting codes

Abstract

A linear binary filter has as its output a binary sequence, each digit of which is the result of a parity check on a selection of preceding output digits and of present and preceding digits of the filter input sequence. The terminal properties of these filters may be described by transfer ratios of polynomials in a delay operator. If two binary filters have transfer ratios which are reciprocally related then the filters are mutually inverse in the sense that, in a cascade connection, the second filter unscrambles the scrambling produced by the first. The coding of a finite sequence of binary information digits for protection against noise may be accomplished by a binary sequence filter, the output of which becomes the sequence to be transmitted. (The inverse filter is utilized at the receiver.) Into the filter at the transmitter is inserted a sequence of information digits, immediately followed by another sequence of completely predictable digits consisting, say, of zeros. The completed block of digits is scrambled in a linear filter before transmission through the noisy channel. If this scrambled sequence were unaffected by noise in the channel the result of unscrambling by the receiver filter would be the original sequence of information digits followed by the all-zero sequence. If, however, channel noise has been added to the sequence put into the receiver filter, then its output is the original sequence, plus the response of the receiver filter to the noise superimposed thereon. In particular, the sequence positions which would have contained all zeros, had there been no noise, will now contain digits whose values are related to the sequence position affected by the noise. These data may then be utilized for the subsequent correction of the errors which would otherwise have been caused by the noise.