MinT - Best Known (87−58, 87, s)-Nets in Base 3

Best Known (87−58, 87, s)-Nets in Base 3

(87−58, 87, 37)-Net over F₃ — Constructive and digital

Digital (29, 87, 37)-net over F₃, using

t-expansion [i] based on digital (27, 87, 37)-net over F₃, using
- net from sequence [i] based on digital (27, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(87−58, 87, 42)-Net over F₃ — Digital

Digital (29, 87, 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]

(87−58, 87, 97)-Net in Base 3 — Upper bound on s

There is no (29, 87, 98)-net in base 3, because

1 times m-reduction [i] would yield (29, 86, 98)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁸⁶, 98, S₃, 57), but
  - the linear programming bound shows that MÂ â‰¥ 124033 090266 094481 284342 649580 843772 229329 031739 / 790801 > 3⁸⁶ [i]