MinT - Best Known (80, 80+161, s)-Nets in Base 3

Best Known (80, 80+161, s)-Nets in Base 3

(80, 80+161, 55)-Net over F₃ — Constructive and digital

Digital (80, 241, 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, 80+161, 84)-Net over F₃ — Digital

Digital (80, 241, 84)-net over F₃, using

t-expansion [i] based on digital (71, 241, 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, 80+161, 251)-Net over F₃ — Upper bound on s (digital)

There is no digital (80, 241, 252)-net over F₃, because

2 times m-reduction [i] would yield digital (80, 239, 252)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁹, 252, F₃, 159) (dual of [252, 13, 160]-code), but
  - residual code [i] would yield OA(3⁸⁰, 92, S₃, 53), but
    - the linear programming bound shows that MÂ â‰¥ 2 550145 733885 710214 972383 625690 731033 510053 / 13237 > 3⁸⁰ [i]

(80, 80+161, 339)-Net in Base 3 — Upper bound on s

There is no (80, 241, 340)-net in base 3, because

1 times m-reduction [i] would yield (80, 240, 340)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 675890 160624 077296 228090 887535 243749 915757 017218 390385 546460 362642 068011 065178 484304 627397 932491 474827 210274 950785 > 3²⁴⁰ [i]