MinT - Best Known (81, 81+155, s)-Nets in Base 3

Best Known (81, 81+155, s)-Nets in Base 3

(81, 81+155, 56)-Net over F₃ — Constructive and digital

Digital (81, 236, 56)-net over F₃, using

net from sequence [i] based on digital (81, 55)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 55)-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]

(81, 81+155, 84)-Net over F₃ — Digital

Digital (81, 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]

(81, 81+155, 266)-Net over F₃ — Upper bound on s (digital)

There is no digital (81, 236, 267)-net over F₃, because

2 times m-reduction [i] would yield digital (81, 234, 267)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁴, 267, F₃, 153) (dual of [267, 33, 154]-code), but
  - residual code [i] would yield OA(3⁸¹, 113, S₃, 51), but
    - the linear programming bound shows that MÂ â‰¥ 23713 080736 201971 210054 022630 767751 231756 211462 816760 345549 / 44 946009 615547 625000 > 3⁸¹ [i]

(81, 81+155, 350)-Net in Base 3 — Upper bound on s

There is no (81, 236, 351)-net in base 3, because

1 times m-reduction [i] would yield (81, 235, 351)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15330 027230 855250 222328 336065 830407 571587 593219 181973 461457 115065 909888 374538 261772 424192 379872 620730 380867 966759 > 3²³⁵ [i]