MinT - Best Known (227−144, 227, s)-Nets in Base 3

Best Known (227−144, 227, s)-Nets in Base 3

(227−144, 227, 58)-Net over F₃ — Constructive and digital

Digital (83, 227, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(227−144, 227, 84)-Net over F₃ — Digital

Digital (83, 227, 84)-net over F₃, using

t-expansion [i] based on digital (71, 227, 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]

(227−144, 227, 368)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 227, 369)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²²⁷, 369, F₃, 144) (dual of [369, 142, 145]-code), but
- residual code [i] would yield linear OA(3⁸³, 224, F₃, 48) (dual of [224, 141, 49]-code), but
  - the Johnson bound shows that NÂ â‰¤ 18 650275 231410 181575 562142 123676 906294 083572 717553 183697 141026 684466 < 3¹⁴¹ [i]

(227−144, 227, 374)-Net in Base 3 — Upper bound on s

There is no (83, 227, 375)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 304383 703732 961449 492908 206202 051460 215054 609000 892184 941402 833997 703852 556524 932449 207019 959124 291437 984241 > 3²²⁷ [i]