MinT - Best Known (20, ∞, s)-Nets in Base 5

Best Known (20, ∞, s)-Nets in Base 5

(20, ∞, 43)-Net over F₅ — Constructive and digital

Digital (20, m, 43)-net over F₅ for arbitrarily large m, using

net from sequence [i] based on digital (20, 42)-sequence over F₅, using
- t-expansion [i] based on digital (18, 42)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 17, N(F)Â =Â 42, and 1 place with degree 2 [i] based on function field F/F₅ with g(F) = 17 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(20, ∞, 45)-Net over F₅ — Digital

Digital (20, m, 45)-net over F₅ for arbitrarily large m, using

net from sequence [i] based on digital (20, 44)-sequence over F₅, using
- t-expansion [i] based on digital (19, 44)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 19 and N(F) ≥ 45, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(20, ∞, 94)-Net in Base 5 — Upper bound on s

There is no (20, m, 95)-net in base 5 for arbitrarily large m, because

m-reduction [i] would yield (20, 281, 95)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5²⁸¹, 95, S₅, 3, 261), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 4 504112 890784 525303 371583 151787 727847 532332 133580 294973 788106 761215 637819 628193 185165 565648 995788 215542 125035 645294 759063 345617 996739 121292 201506 081179 270403 296033 276063 781158 882193 267345 428466 796875 / 131 > 5²⁸¹ [i]