MinT - Best Known (84, 84+158, s)-Nets in Base 3

Best Known (84, 84+158, s)-Nets in Base 3

(84, 84+158, 59)-Net over F₃ — Constructive and digital

Digital (84, 242, 59)-net over F₃, using

net from sequence [i] based on digital (84, 58)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 58)-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]

(84, 84+158, 84)-Net over F₃ — Digital

Digital (84, 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]

(84, 84+158, 283)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 242, 284)-net over F₃, because

2 times m-reduction [i] would yield digital (84, 240, 284)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁰, 284, F₃, 156) (dual of [284, 44, 157]-code), but
  - residual code [i] would yield OA(3⁸⁴, 127, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 75093 992178 172944 979964 312324 767811 438315 782053 828257 111860 508367 900579 / 6 152332 044204 270968 974779 651985 > 3⁸⁴ [i]

(84, 84+158, 364)-Net in Base 3 — Upper bound on s

There is no (84, 242, 365)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 33 009061 878482 809999 129596 681656 047420 392089 467966 527050 561687 620948 651688 770692 628358 636800 613142 484880 496196 026651 > 3²⁴² [i]