MinT - Best Known (29, ∞, s)-Nets in Base 3

Best Known (29, ∞, s)-Nets in Base 3

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

Digital (29, m, 37)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (29, 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]

(29, ∞, 42)-Net over F₃ — Digital

Digital (29, m, 42)-net over F₃ for arbitrarily large m, using

net from sequence [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]

(29, ∞, 68)-Net in Base 3 — Upper bound on s

There is no (29, m, 69)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (29, 271, 69)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁷¹, 69, S₃, 4, 242), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 2142 368937 253103 043834 638591 707247 385061 383828 483530 475078 497325 497761 213758 635430 004735 714829 845784 822552 287049 510582 282468 757069 > 3²⁷¹ [i]