No abstract

Let ÀDo À 4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of Qð ffiffiffiffiffiffiffi ffi ÀD p Þ exists. Let d be a fundamental discriminant prime to D: Let 2k À 1 be an odd natural number prime to the class number of Qð ffiffiffiffiffiffiffi ffi ÀD p Þ: Let w be the twist of the ð2k À 1Þth power of a canonical Hecke character of Qð ffiffiffiffiffiffiffi ffi ÀD p Þ by the Kronecker's symbol n/ð d n Þ: It is proved that the vanishing order of the Hecke L-function Lðs; wÞ at its central point s ¼ k is determined by its root number when jdj5D 1 12 ÀE ; where the constant implied in the symbol 5 depends only on k and E; and is effective for L-functions with root number À1: r

No abstract

The lower bounds for the translative kissing numbers of superballs are studied in this note. We improve the bound given by Larman and Zong. Furthermore, we give a constructive bound based on algebraic-geometry codes that also improves the bound by Larman and Zong in almost all cases.

In the present paper, we focus on constructive spherical codes. By employing algebraic geometry codes, we give an explicit construction of spherical code sequences. By making use of the idea involved in the proof of the Gilbert-Varshamov bound in coding theory, we construct a spherical code sequence in exponential time which meets the best-known asymptotic bound by Shamsiev and Wyner.