MinT - Best Known (84, 84+144, s)-Nets in Base 3

Best Known (84, 84+144, s)-Nets in Base 3

(84, 84+144, 59)-Net over F₃ — Constructive and digital

Digital (84, 228, 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+144, 84)-Net over F₃ — Digital

Digital (84, 228, 84)-net over F₃, using

t-expansion [i] based on digital (71, 228, 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+144, 379)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 228, 380)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²²⁸, 380, F₃, 144) (dual of [380, 152, 145]-code), but
- residual code [i] would yield linear OA(3⁸⁴, 235, F₃, 48) (dual of [235, 151, 49]-code), but
  - the Johnson bound shows that NÂ â‰¤ 1 014891 606187 829584 057582 115648 929135 604017 524062 355605 507975 062077 667701 < 3¹⁵¹ [i]

(84, 84+144, 380)-Net in Base 3 — Upper bound on s

There is no (84, 228, 381)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 137034 488393 716759 109164 304629 105727 428752 388533 651703 856325 752367 755672 958602 569999 477617 439352 079754 903137 > 3²²⁸ [i]