MinT - Best Known (84, 84+152, s)-Nets in Base 3

Best Known (84, 84+152, s)-Nets in Base 3

(84, 84+152, 59)-Net over F₃ — Constructive and digital

Digital (84, 236, 59)-net over F₃, using

net from sequence [i] based on digital (84, 58)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 58)-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]

(84, 84+152, 84)-Net over F₃ — Digital

Digital (84, 236, 84)-net over F₃, using

t-expansion [i] based on digital (71, 236, 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]

(84, 84+152, 361)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 236, 362)-net over F₃, because

2 times m-reduction [i] would yield digital (84, 234, 362)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁴, 362, F₃, 150) (dual of [362, 128, 151]-code), but
  - residual code [i] would yield linear OA(3⁸⁴, 211, F₃, 50) (dual of [211, 127, 51]-code), but
    - the Johnson bound shows that NÂ â‰¤ 3 465392 580935 096296 712977 980638 603324 843243 174078 085715 358657 < 3¹²⁷ [i]

(84, 84+152, 370)-Net in Base 3 — Upper bound on s

There is no (84, 236, 371)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 42016 842738 016132 342746 982998 395015 241445 330521 365272 593434 971435 351609 980430 360151 172640 723352 391363 176585 060553 > 3²³⁶ [i]