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Minimal redundant digit expansions in the gaussian integers

Clemens Heuberger — 2002

Journal de théorie des nombres de Bordeaux

We consider minimal redundant digit expansions in canonical number systems in the gaussian integers. In contrast to the case of rational integers, where the knowledge of the two least significant digits in the “standard” expansion suffices to calculate the least significant digit in a minimal redundant expansion, such a property does not hold in the gaussian numbers : We prove that there exist pairs of numbers whose non-redundant expansions agree arbitrarily well but which have different least significant...

Optimality of the Width- w Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Clemens HeubergerDaniel Krenn — 2013

Journal de Théorie des Nombres de Bordeaux

We consider digit expansions j = 0 - 1 Φ j ( d j ) with an endomorphism Φ of an Abelian group. In such a numeral system, the w -NAF condition (each block of w consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight 1 admits an optimal w -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar multiplication...

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