MinT - Best Known (80, 237, s)-Nets in Base 3

Best Known (80, 237, s)-Nets in Base 3

(80, 237, 55)-Net over F₃ — Constructive and digital

Digital (80, 237, 55)-net over F₃, using

net from sequence [i] based on digital (80, 54)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 54)-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]

(80, 237, 84)-Net over F₃ — Digital

Digital (80, 237, 84)-net over F₃, using

t-expansion [i] based on digital (71, 237, 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]

(80, 237, 254)-Net over F₃ — Upper bound on s (digital)

There is no digital (80, 237, 255)-net over F₃, because

1 times m-reduction [i] would yield digital (80, 236, 255)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁶, 255, F₃, 156) (dual of [255, 19, 157]-code), but
  - residual code [i] would yield OA(3⁸⁰, 98, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 341 054785 910915 827613 087262 939178 024688 921005 310349 / 1 990980 656333 > 3⁸⁰ [i]

(80, 237, 342)-Net in Base 3 — Upper bound on s

There is no (80, 237, 343)-net in base 3, because

1 times m-reduction [i] would yield (80, 236, 343)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 43248 894770 745101 005129 374173 288627 567959 478093 314797 125863 764099 484380 860408 956402 490524 256085 932931 297825 186645 > 3²³⁶ [i]