MinT - Best Known (30, ∞, s)-Nets in Base 3

Best Known (30, ∞, s)-Nets in Base 3

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

Digital (30, m, 37)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (30, 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]

(30, ∞, 42)-Net over F₃ — Digital

Digital (30, m, 42)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (30, 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]

(30, ∞, 70)-Net in Base 3 — Upper bound on s

There is no (30, m, 71)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (30, 279, 71)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁷⁹, 71, S₃, 4, 249), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1688 183989 188180 427341 259972 967797 381905 844688 872551 190547 111826 823234 339298 187164 722665 875174 834040 361066 428593 820517 049221 778799 888843 / 125 > 3²⁷⁹ [i]