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Published January 12, 2006 | public
Book Section - Chapter

On Generalized Parity Checks

Abstract

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.

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.

Additional details

Created:
August 22, 2023
Modified:
January 14, 2024