MinT - Best Known (26, ∞, s)-Nets in Base 3

Best Known (26, ∞, s)-Nets in Base 3

(26, ∞, 36)-Net over F₃ — Constructive and digital

Digital (26, m, 36)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (26, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
  - an explicitly constructive algebraic function field [i]

(26, ∞, 37)-Net over F₃ — Digital

Digital (26, m, 37)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (26, 36)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 25, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(26, ∞, 62)-Net in Base 3 — Upper bound on s

There is no (26, m, 63)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (26, 247, 63)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁴⁷, 63, S₃, 4, 221), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 317806 246861 327692 649610 852285 462512 835434 043915 247229 061739 424851 477075 056621 938593 968379 410401 484960 703396 297551 210415 / 37 > 3²⁴⁷ [i]