MinT - Best Known (85−53, 85, s)-Nets in Base 3

Best Known (85−53, 85, s)-Nets in Base 3

(85−53, 85, 38)-Net over F₃ — Constructive and digital

Digital (32, 85, 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]

(85−53, 85, 42)-Net over F₃ — Digital

Digital (32, 85, 42)-net over F₃, using

t-expansion [i] based on digital (29, 85, 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]

(85−53, 85, 122)-Net in Base 3 — Upper bound on s

There is no (32, 85, 123)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁵, 123, S₃, 53), but
- the linear programming bound shows that MÂ â‰¥ 18 556558 104235 515171 973211 626345 894318 538602 254870 483785 952441 / 483 199328 091023 596855 > 3⁸⁵ [i]