p-adic Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound

TL;DR

The paper extends upper-bound methods for spherical codes to p-adic Hilbert spaces by defining p-adic spherical codes in $\mathbb{Q}_p^d$ and proving a p-adic Pfender-type bound. It adapts the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender framework to the non-Archimedean setting via a nonnegativity condition on a sum of a function of inner-product-derived distances, yielding a concrete bound on $n$. A special p-adic kissing-number bound at $\theta=\pi/3$ is obtained, mirroring the classical bound but in the $p$-adic context. These results broaden the coding-theory toolbox to non-Archimedean spaces and open avenues for studying p-adic kissing numbers and related extremal problems.

Abstract

We introduce the notion of p-adic spherical codes (in particular, p-adic kissing number problem). We show that the one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes, obtained by Pfender \textit{[J. Combin. Theory Ser. A, 2007]}, extends to p-adic Hilbert spaces.

Theorems & Definitions (9)

• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Example 2.2
• Definition 2.3
• Theorem 2.4
• proof
• Corollary 2.5
• Theorem 2.6