MinT - Best Known (250−166, 250, s)-Nets in Base 3

Best Known (250−166, 250, s)-Nets in Base 3

(250−166, 250, 59)-Net over F₃ — Constructive and digital

Digital (84, 250, 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]

(250−166, 250, 84)-Net over F₃ — Digital

Digital (84, 250, 84)-net over F₃, using

t-expansion [i] based on digital (71, 250, 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]

(250−166, 250, 264)-Net over F₃ — Upper bound on s (digital)

There is no digital (84, 250, 265)-net over F₃, because

1 times m-reduction [i] would yield digital (84, 249, 265)-net over F₃, but
- 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]

(250−166, 250, 357)-Net in Base 3 — Upper bound on s

There is no (84, 250, 358)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 223561 617269 563835 876730 634913 362520 604294 325161 835789 419668 366956 657529 870888 413264 098764 416513 069502 934201 304652 154945 > 3²⁵⁰ [i]