MinT - Best Known (28, ∞, s)-Nets in Base 3

Best Known (28, ∞, s)-Nets in Base 3

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

Digital (28, m, 37)-net over F₃ for arbitrarily large m, using

net from sequence [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]

(28, ∞, 39)-Net over F₃ — Digital

Digital (28, m, 39)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (28, 38)-sequence over F₃, using
- t-expansion [i] based on digital (27, 38)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 27 and N(F) ≥ 39, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(28, ∞, 66)-Net in Base 3 — Upper bound on s

There is no (28, m, 67)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (28, 263, 67)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁶³, 67, S₃, 4, 235), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 20 064758 033021 825016 756938 704793 291814 112577 363404 712835 984223 567880 266795 525746 896596 077363 242489 257597 339427 453909 242321 972182 / 59 > 3²⁶³ [i]