MinT - Best Known (237−150, 237, s)-Nets in Base 3

Best Known (237−150, 237, s)-Nets in Base 3

(237−150, 237, 62)-Net over F₃ — Constructive and digital

Digital (87, 237, 62)-net over F₃, using

net from sequence [i] based on digital (87, 61)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 61)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(237−150, 237, 84)-Net over F₃ — Digital

Digital (87, 237, 84)-net over F₃, using

t-expansion [i] based on digital (71, 237, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(237−150, 237, 388)-Net over F₃ — Upper bound on s (digital)

There is no digital (87, 237, 389)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³⁷, 389, F₃, 150) (dual of [389, 152, 151]-code), but
- residual code [i] would yield linear OA(3⁸⁷, 238, F₃, 50) (dual of [238, 151, 51]-code), but
  - the Johnson bound shows that NÂ â‰¤ 1 109935 900942 577097 815120 032765 202024 203565 946556 215381 480293 664347 835582 < 3¹⁵¹ [i]

(237−150, 237, 392)-Net in Base 3 — Upper bound on s

There is no (87, 237, 393)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 124341 759627 310773 380350 050974 617488 932902 880018 389025 714212 900839 230075 746417 790566 799579 388074 811022 433677 285467 > 3²³⁷ [i]