MinT - Best Known (230−147, 230, s)-Nets in Base 3

Best Known (230−147, 230, s)-Nets in Base 3

(230−147, 230, 58)-Net over F₃ — Constructive and digital

Digital (83, 230, 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]

(230−147, 230, 84)-Net over F₃ — Digital

Digital (83, 230, 84)-net over F₃, using

t-expansion [i] based on digital (71, 230, 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]

(230−147, 230, 363)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 230, 364)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³⁰, 364, F₃, 147) (dual of [364, 134, 148]-code), but
- residual code [i] would yield linear OA(3⁸³, 216, F₃, 49) (dual of [216, 133, 50]-code), but
  - 1 times truncation [i] would yield linear OA(3⁸², 215, F₃, 48) (dual of [215, 133, 49]-code), but
    - the Johnson bound shows that NÂ â‰¤ 2633 391490 584373 459265 026071 777086 821369 449862 527703 301746 510338 < 3¹³³ [i]

(230−147, 230, 371)-Net in Base 3 — Upper bound on s

There is no (83, 230, 372)-net in base 3, because

1 times m-reduction [i] would yield (83, 229, 372)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19 584202 707791 189739 576813 154645 097826 957024 657042 963745 591293 814617 808301 676539 714168 058047 569873 160727 743721 > 3²²⁹ [i]