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On Generalized Parity Checks

McEliece, Robert J. and Soedarmadji, Edwin (2006) On Generalized Parity Checks. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science. No.3857. Springer , Berlin, pp. 29-34. ISBN 978-3-540-31423-3.

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An ordinary parity-check is an extra bit p appended to a block (x₁, ..., x_k) of k information bits such that the resulting codeword (x₁, ..., x_k ,p) is capable of detecting one error. The choices for p are p₀  =  x₁ + ... + x_k (mod 2) (even parity) p₁  =  x₁  + ... + x_k + 1 (mod 2) (odd parity) In this paper we consider defining a parity-check if the underlying alphabet is nonbinary. The obvious definition is of course p  =  x₁  + ... + x_k  +  α(mod q). We shall show that this obvious choice is the only choice for q = 2, and up to a natural equivalence the only choice for q = 3. For q ≥ 4, however, the situation is much more complicated.

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Additional Information:© 2006 Springer-Verlag Berlin Heidelberg. This work was sponsored by NSF grant CCF-0514881, Qualcomm, Sony, and the Lee Center for Advanced Networking.
Funding AgencyGrant Number
Sony CorporationUNSPECIFIED
Caltech Lee Center for Advanced NetworkingUNSPECIFIED
Subject Keywords:Unit Ball; Cyclic Group; Parity Check; Obvious Choice; Gray Code
Series Name:Lecture Notes in Computer Science
Issue or Number:3857
Record Number:CaltechAUTHORS:20200204-111546794
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101114
Deposited By: Tony Diaz
Deposited On:04 Feb 2020 19:25
Last Modified:16 Nov 2021 17:59

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