Weak radix representation and cyclic codes over euclidean domains

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Abstract
Here we show that to some extent the two theories cyclic arithmetic coding and cyclic polynomial coding can be subsumed under a “unified” theory based on a more or less arbitrary Euclidean domain. In particular, we show that Hamming weight (for cyclic polynomial codes) and arithmetic weight (for cyclic arithmetic codes) are special cases of a more general notion of weight defined for Euclidean rings. of fundamental importance in this development is a general-ization of the notion of radix representation for integers which we call a weak radix representation. For several specific Euclidean rings we prove the existence of certain unique canonical weak radix representations for all ring elements. We show, for example, that every Gaussian integer has a unique representation of the form (1) each aj is zero or a unit, and whenever aj≠0 , then aj+1 = aj+2 = 0 This is analogous to a known result of Reitwiesner that each rational integer has a unique representation of the form (1) where r - 2, each aj. is 0 or ± 1, and whenever aj ≠ 0, aj+1 = 0.

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