MinT - Best Known (106, 106+139, s)-Nets in Base 3

Best Known (106, 106+139, s)-Nets in Base 3

(106, 106+139, 73)-Net over F₃ — Constructive and digital

Digital (106, 245, 73)-net over F₃, using

net from sequence [i] based on digital (106, 72)-sequence over F₃, using
- base reduction for sequences [i] based on digital (17, 72)-sequence over F₉, using
  - s-reduction based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(106, 106+139, 104)-Net over F₃ — Digital

Digital (106, 245, 104)-net over F₃, using

t-expansion [i] based on digital (102, 245, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(106, 106+139, 579)-Net in Base 3 — Upper bound on s

There is no (106, 245, 580)-net in base 3, because

1 times m-reduction [i] would yield (106, 244, 580)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 267 315562 271447 977774 320967 167761 795407 820733 672424 753956 271829 201769 656954 525317 358120 639522 880619 742469 740239 716969 > 3²⁴⁴ [i]