MinT - Best Known (237−162, 237, s)-Nets in Base 3

Best Known (237−162, 237, s)-Nets in Base 3

(237−162, 237, 50)-Net over F₃ — Constructive and digital

Digital (75, 237, 50)-net over F₃, using

net from sequence [i] based on digital (75, 49)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 49)-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]

(237−162, 237, 84)-Net over F₃ — Digital

Digital (75, 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]

(237−162, 237, 234)-Net over F₃ — Upper bound on s (digital)

There is no digital (75, 237, 235)-net over F₃, because

9 times m-reduction [i] would yield digital (75, 228, 235)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁸, 235, F₃, 153) (dual of [235, 7, 154]-code), but
  - residual code [i] would yield linear OA(3⁷⁵, 81, F₃, 51) (dual of [81, 6, 52]-code), but
    - â€œMaâ€ bound on codes from Brouwerâ€™s database [i]

(237−162, 237, 238)-Net in Base 3 — Upper bound on s

There is no (75, 237, 239)-net in base 3, because

3 times m-reduction [i] would yield (75, 234, 239)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²³⁴, 239, S₃, 159), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 358805 113110 950445 279469 426279 262434 966930 339114 122428 335494 730197 493691 743717 859849 693621 015768 243278 975535 908089 / 80 > 3²³⁴ [i]