LetGbe the set of all linear recurring sequences generated by a polynomialf. It is shown that the linear functional advancing the phase of every sequence inGbykmay be obtained essentially as thekth state of a certain member ofGdepending only onfwhen and only whenfis a binomial or trinomial.

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article Free Access Share on On Mappings for Modular Arithmetic, I Authors: T. R. N. Rao Bell Telephone Laboratories, Inc., Holmdel, New Jersey Bell Telephone Laboratories, Inc., Holmdel, New JerseyView Profile , N. Zierler Institute for Defense Analysis, Princeton, N. J. and The MITRE Corp., Bedford, Massachusetts Institute for Defense Analysis, Princeton, N. J. and The MITRE Corp., Bedford, MassachusettsView Profile Authors Info & Claims Journal of the ACMVolume 12Issue 4Oct. 1965 pp 542–544https://doi.org/10.1145/321296.321303Published:01 October 1965Publication History 1citation295DownloadsMetricsTotal Citations1Total Downloads295Last 12 Months8Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

A technique is described for finding simply computable numerical-valued functions of a received binary word whose value indicates where errors in transmission have occurred. Although it seems that a certain condition must usually be fulfilled for such functions to exist, or for our method to constitute an efficient procedure for finding them, there is, on the one hand, a strong tendency for "good" codes to satisfy the condition, while, on the other, it appears to be straightforward to construct codes which are good for a specified channel and also fulfill the condition. An advantage of the resulting decoding procedure is that it corrects and detects all possible errors; more precisely, if a worduis received and the coset\bar{u}to whichubelongs has a unique leadere, the procedure concludes thatu + ewas sent, while ifuhas no unique leader, that fact, along with the weight of\bar{u}(and sometimes a little more) can be indicated. The ideas and techniques are illustrated by the construction of decoding procedures for the perfect (23, 12) three-error-correcting code.