MinT - Best Known (84−52, 84, s)-Nets in Base 3

Best Known (84−52, 84, s)-Nets in Base 3

(84−52, 84, 38)-Net over F₃ — Constructive and digital

Digital (32, 84, 38)-net over F₃, 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]

(84−52, 84, 42)-Net over F₃ — Digital

Digital (32, 84, 42)-net over F₃, using

t-expansion [i] based on digital (29, 84, 42)-net over F₃, 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]

(84−52, 84, 126)-Net in Base 3 — Upper bound on s

There is no (32, 84, 127)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁴, 127, S₃, 52), but
- the linear programming bound shows that MÂ â‰¥ 75093 992178 172944 979964 312324 767811 438315 782053 828257 111860 508367 900579 / 6 152332 044204 270968 974779 651985 > 3⁸⁴ [i]