MinT - Best Known (31, ∞, s)-Nets in Base 3

Best Known (31, ∞, s)-Nets in Base 3

(31, ∞, 37)-Net over F₃ — Constructive and digital

Digital (31, m, 37)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (31, 36)-sequence over F₃, using
- t-expansion [i] based on digital (28, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 28 and N(F) ≥ 37, using
    - an explicitly constructive algebraic function field [i]

(31, ∞, 42)-Net over F₃ — Digital

Digital (31, m, 42)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (31, 41)-sequence over F₃, using
- t-expansion [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(31, ∞, 73)-Net in Base 3 — Upper bound on s

There is no (31, m, 74)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (31, 290, 74)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁹⁰, 74, S₃, 4, 259), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 229 508652 590225 900915 193766 377529 564949 108466 983495 168600 256377 208056 541526 061551 978631 651838 527412 783227 454480 502657 288667 678716 723464 947451 / 65 > 3²⁹⁰ [i]