MinT - Best Known (81, 81+150, s)-Nets in Base 3

Best Known (81, 81+150, s)-Nets in Base 3

(81, 81+150, 56)-Net over F₃ — Constructive and digital

Digital (81, 231, 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+150, 84)-Net over F₃ — Digital

Digital (81, 231, 84)-net over F₃, using

t-expansion [i] based on digital (71, 231, 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+150, 277)-Net over F₃ — Upper bound on s (digital)

There is no digital (81, 231, 278)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²³¹, 278, F₃, 150) (dual of [278, 47, 151]-code), but
- residual code [i] would yield OA(3⁸¹, 127, S₃, 50), but
  - the linear programming bound shows that MÂ â‰¥ 5 298495 360699 143400 851017 463794 435462 430841 716143 016814 949397 477824 916043 / 11181 941894 946610 416397 751274 583291 > 3⁸¹ [i]

(81, 81+150, 354)-Net in Base 3 — Upper bound on s

There is no (81, 231, 355)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 188 548287 831583 169465 296029 200349 778021 879550 302733 893587 732770 191599 673024 586292 358323 543349 050297 809896 775451 > 3²³¹ [i]