MinT - Best Known (32, ∞, s)-Nets in Base 3

Best Known (32, ∞, s)-Nets in Base 3

(32, ∞, 38)-Net over F₃ — Constructive and digital

Digital (32, m, 38)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (32, 37)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
  - an explicitly constructive algebraic function field [i]

(32, ∞, 42)-Net over F₃ — Digital

Digital (32, m, 42)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (32, 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]

(32, ∞, 75)-Net in Base 3 — Upper bound on s

There is no (32, m, 76)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (32, 298, 76)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²⁹⁸, 76, S₃, 4, 266), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 962111 199839 766722 542339 725809 932528 852646 303963 169922 757882 724456 622292 877486 764511 136288 231541 493231 590276 981904 581280 327015 836308 306102 112681 / 89 > 3²⁹⁸ [i]