MinT - Best Known (15, ∞, s)-Nets in Base 5

Best Known (15, ∞, s)-Nets in Base 5

(15, ∞, 36)-Net over F₅ — Constructive and digital

Digital (15, m, 36)-net over F₅ for arbitrarily large m, using

net from sequence [i] based on digital (15, 35)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 11, N(F)Â =Â 32, and 4 places with degree 2 [i] based on function field F/F₅ with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(15, ∞, 39)-Net over F₅ — Digital

Digital (15, m, 39)-net over F₅ for arbitrarily large m, using

net from sequence [i] based on digital (15, 38)-sequence over F₅, using
- t-expansion [i] based on digital (14, 38)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 14 and N(F) ≥ 39, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(15, ∞, 73)-Net in Base 5 — Upper bound on s

There is no (15, m, 74)-net in base 5 for arbitrarily large m, because

m-reduction [i] would yield (15, 218, 74)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5²¹⁸, 74, S₅, 3, 203), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 13056 405650 419723 277775 708120 647691 087053 828021 227424 328634 697487 389886 317136 615343 670538 531185 693700 127912 995358 724079 775214 477194 822393 357753 753662 109375 / 51 > 5²¹⁸ [i]