MinT - Best Known (84, 84+165, s)-Nets in Base 3

Best Known (84, 84+165, s)-Nets in Base 3

(84, 84+165, 59)-Net over F₃ — Constructive and digital

Digital (84, 249, 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+165, 84)-Net over F₃ — Digital

Digital (84, 249, 84)-net over F₃, using

t-expansion [i] based on digital (71, 249, 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+165, 264)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 249, 265)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁹, 265, F₃, 165) (dual of [265, 16, 166]-code), but
- residual code [i] would yield OA(3⁸⁴, 99, S₃, 55), but
  - the linear programming bound shows that MÂ â‰¥ 4 602106 814071 993271 061293 737170 927222 039075 353067 / 326 668160 > 3⁸⁴ [i]

(84, 84+165, 358)-Net in Base 3 — Upper bound on s

There is no (84, 249, 359)-net in base 3, because

1 times m-reduction [i] would yield (84, 248, 359)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21911 273496 817928 706238 558548 639408 008598 294103 993761 047142 381861 689119 083591 436102 550129 705112 024427 594538 551658 619565 > 3²⁴⁸ [i]