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• W2029206768 abstract "The theory of cyclotomy dates back to Gauss and has a number of applications in combinatorics, coding theory, and cryptography. Cyclotomy over a residue class ring <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>${BBZ}_{v}$</tex></formula> can be divided into classical cyclotomy or generalized cyclotomy, depending on <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$v$</tex> </formula> prime or composite. In this paper, we introduce a generalized cyclotomy of order <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$d$</tex></formula> over <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex Notation=TeX>${BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},ldots, p_{n}^{e_{n}}}$</tex></formula> , which includes Whiteman's and Ding-Helleseth's generalized cyclotomy as special cases. Here, <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex Notation=TeX>$p_{1},p_{2},ldots,p_{n}$</tex></formula> are pairwise distinct odd primes satisfying <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$dvert (p_{i}-1)$</tex></formula> for all <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$1leq ileq n$</tex></formula> and <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>$e_{1},e_{2},ldots,e_{n}$</tex></formula> are positive integers. We derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers. As applications, we completely solve an open problem and a conjecture on Whiteman's generalized cyclotomy of order four over <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>${BBZ}_{p_{1}p_{2}}$</tex> </formula> . Besides, we also construct an infinite series of near-optimal codebooks over <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex Notation=TeX>${BBZ}_{p_{1}p_{2}}$</tex></formula> , as well as some infinite series of asymptotically optimal difference systems of sets over <formula formulatype=inline xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink><tex Notation=TeX>${BBZ}_{p_{1}^{e_{1}}p_{2}^{e_{2}},ldots,p_{n}^{e_{n}}}$</tex> </formula> ." @default.
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• W2029206768 date "2014-02-01" @default.
• W2029206768 modified "2023-10-14" @default.
• W2029206768 title "A Unified Approach to Whiteman's and Ding-Helleseth's Generalized Cyclotomy Over Residue Class Rings" @default.
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