MinT - Best Known (235−155, 235, s)-Nets in Base 3

Best Known (235−155, 235, s)-Nets in Base 3

(235−155, 235, 55)-Net over F₃ — Constructive and digital

Digital (80, 235, 55)-net over F₃, using

net from sequence [i] based on digital (80, 54)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 54)-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]

(235−155, 235, 84)-Net over F₃ — Digital

Digital (80, 235, 84)-net over F₃, using

t-expansion [i] based on digital (71, 235, 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]

(235−155, 235, 258)-Net over F₃ — Upper bound on s (digital)

There is no digital (80, 235, 259)-net over F₃, because

2 times m-reduction [i] would yield digital (80, 233, 259)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³³, 259, F₃, 153) (dual of [259, 26, 154]-code), but
  - residual code [i] would yield OA(3⁸⁰, 105, S₃, 51), but
    - the linear programming bound shows that MÂ â‰¥ 19 616630 525054 787027 648895 210370 192777 110899 155889 / 122567 513497 > 3⁸⁰ [i]

(235−155, 235, 344)-Net in Base 3 — Upper bound on s

There is no (80, 235, 345)-net in base 3, because

1 times m-reduction [i] would yield (80, 234, 345)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5027 054501 914091 044047 052928 614955 605427 119967 519714 557545 627608 392844 014893 191463 710760 300405 385190 280012 235395 > 3²³⁴ [i]