MinT - Best Known (84, 84+164, s)-Nets in Base 3

Best Known (84, 84+164, s)-Nets in Base 3

(84, 84+164, 59)-Net over F₃ — Constructive and digital

Digital (84, 248, 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+164, 84)-Net over F₃ — Digital

Digital (84, 248, 84)-net over F₃, using

t-expansion [i] based on digital (71, 248, 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+164, 268)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 248, 269)-net over F₃, because

2 times m-reduction [i] would yield digital (84, 246, 269)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁶, 269, F₃, 162) (dual of [269, 23, 163]-code), but
  - residual code [i] would yield OA(3⁸⁴, 106, S₃, 54), but
    - the linear programming bound shows that MÂ â‰¥ 554636 234644 780595 906506 938993 021399 906097 624808 134107 / 34 652730 233275 > 3⁸⁴ [i]

(84, 84+164, 358)-Net in Base 3 — Upper bound on s

There is no (84, 248, 359)-net in base 3, because

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]