New nonbinary code bounds based on divisibility arguments
| Autor: | Sven Polak |
|---|---|
| Přispěvatelé: | Algebra, Geometry & Mathematical Physics (KDV, FNWI) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Upper bounds
Divisibility 0102 computer and information sciences 02 engineering and technology Nonbinary code 01 natural sciences Article Combinatorics Mathematics::Group Theory Mathematics::Algebraic Geometry 0202 electrical engineering electronic engineering information engineering Code (cryptography) FOS: Mathematics Mathematics - Combinatorics Maximum size Mathematics Kirkman system Code Applied Mathematics Minimum distance 020206 networking & telecommunications Divisibility rule 16. Peace & justice Net (mathematics) Computer Science Applications 010201 computation theory & mathematics 94B65 05B30 94B65 05B30 Combinatorics (math.CO) Symmetric net |
| Zdroj: | Designs, Codes and Cryptography, 86, 861-874 Designs, Codes, and Cryptography, 86(4). Springer Netherlands Designs, Codes, and Cryptography |
| ISSN: | 0925-1022 |
| Popis: | For $q,n,d \in \mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \leq 65$, $A_4(11,8)\leq 60$ and $A_3(16,11) \leq 29$. These in turn imply the new upper bounds $A_5(9,6) \leq 325$, $A_5(10,6) \leq 1625$, $A_5(11,6) \leq 8125$ and $A_4(12,8) \leq 240$. Furthermore, we prove that for $\mu,q \in \mathbb{N}$, there is a 1-1-correspondence between symmetric $(\mu,q)$-nets (which are certain designs) and codes $C \subseteq [q]^{\mu q}$ of size $\mu q^2$ with minimum distance at least $\mu q - \mu$. We derive the new upper bounds $A_4(9,6) \leq 120$ and $A_4(10,6) \leq 480$ from these `symmetric net' codes. Comment: Revisions have been made based on comments of the referees. 13 pages. To appear in Designs, Codes and Cryptography |
| Databáze: | OpenAIRE |
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