MinT - Best Known (242−165, 242, s)-Nets in Base 3

Best Known (242−165, 242, s)-Nets in Base 3

(242−165, 242, 52)-Net over F₃ — Constructive and digital

Digital (77, 242, 52)-net over F₃, using

net from sequence [i] based on digital (77, 51)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 51)-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]

(242−165, 242, 84)-Net over F₃ — Digital

Digital (77, 242, 84)-net over F₃, using

t-expansion [i] based on digital (71, 242, 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]

(242−165, 242, 241)-Net over F₃ — Upper bound on s (digital)

There is no digital (77, 242, 242)-net over F₃, because

9 times m-reduction [i] would yield digital (77, 233, 242)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³³, 242, F₃, 156) (dual of [242, 9, 157]-code), but
  - residual code [i] would yield OA(3⁷⁷, 85, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 626561 627887 412368 936876 728064 858798 530639 / 100223 > 3⁷⁷ [i]

(242−165, 242, 245)-Net in Base 3 — Upper bound on s

There is no (77, 242, 246)-net in base 3, because

1 times m-reduction [i] would yield (77, 241, 246)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²⁴¹, 246, S₃, 164), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 784 706782 373648 623826 199635 272746 945272 676651 642585 750769 726974 941918 703843 510959 491279 949161 485148 051119 497030 990643 / 55 > 3²⁴¹ [i]