MinT - Best Known (85, 85+164, s)-Nets in Base 3

Best Known (85, 85+164, s)-Nets in Base 3

(85, 85+164, 60)-Net over F₃ — Constructive and digital

Digital (85, 249, 60)-net over F₃, using

net from sequence [i] based on digital (85, 59)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 59)-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]

(85, 85+164, 84)-Net over F₃ — Digital

Digital (85, 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]

(85, 85+164, 273)-Net over F₃ — Upper bound on s (digital)

There is no digital (85, 249, 274)-net over F₃, because

2 times m-reduction [i] would yield digital (85, 247, 274)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁷, 274, F₃, 162) (dual of [274, 27, 163]-code), but
  - residual code [i] would yield OA(3⁸⁵, 111, S₃, 54), but
    - the linear programming bound shows that MÂ â‰¥ 171 512814 450641 798610 864431 673363 023574 527618 893419 348639 / 4028 884726 616750 > 3⁸⁵ [i]

(85, 85+164, 364)-Net in Base 3 — Upper bound on s

There is no (85, 249, 365)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 68344 153945 943613 250955 700416 657173 979643 709508 241423 765516 785097 190285 375609 796823 822011 370533 535749 280524 750910 377377 > 3²⁴⁹ [i]