MinT - Best Known (229−146, 229, s)-Nets in Base 3

Best Known (229−146, 229, s)-Nets in Base 3

(229−146, 229, 58)-Net over F₃ — Constructive and digital

Digital (83, 229, 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]

(229−146, 229, 84)-Net over F₃ — Digital

Digital (83, 229, 84)-net over F₃, using

t-expansion [i] based on digital (71, 229, 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]

(229−146, 229, 368)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 229, 369)-net over F₃, because

2 times m-reduction [i] would yield digital (83, 227, 369)-net over F₃, but
- 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]

(229−146, 229, 371)-Net in Base 3 — Upper bound on s

There is no (83, 229, 372)-net in base 3, because

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]