MinT - Best Known (244−161, 244, s)-Nets in Base 3

Best Known (244−161, 244, s)-Nets in Base 3

(244−161, 244, 58)-Net over F₃ — Constructive and digital

Digital (83, 244, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(244−161, 244, 84)-Net over F₃ — Digital

Digital (83, 244, 84)-net over F₃, using

t-expansion [i] based on digital (71, 244, 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]

(244−161, 244, 267)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 244, 268)-net over F₃, because

2 times m-reduction [i] would yield digital (83, 242, 268)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴², 268, F₃, 159) (dual of [268, 26, 160]-code), but
  - residual code [i] would yield OA(3⁸³, 108, S₃, 53), but
    - the linear programming bound shows that MÂ â‰¥ 351 790187 636131 637637 070482 425686 777069 567336 538130 612039 / 72418 367747 573783 > 3⁸³ [i]

(244−161, 244, 356)-Net in Base 3 — Upper bound on s

There is no (83, 244, 357)-net in base 3, because

1 times m-reduction [i] would yield (83, 243, 357)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 95 800632 147021 715293 477099 813934 152378 840259 248382 539925 384081 418877 384488 272879 759034 936396 691855 950687 432084 698305 > 3²⁴³ [i]