MinT - Best Known (241−123, 241, s)-Nets in Base 3

Best Known (241−123, 241, s)-Nets in Base 3

(241−123, 241, 75)-Net over F₃ — Constructive and digital

Digital (118, 241, 75)-net over F₃, using

net from sequence [i] based on digital (118, 74)-sequence over F₃, using
- base reduction for sequences [i] based on digital (22, 74)-sequence over F₉, using
  - s-reduction based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(241−123, 241, 120)-Net over F₃ — Digital

Digital (118, 241, 120)-net over F₃, using

t-expansion [i] based on digital (113, 241, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(241−123, 241, 829)-Net in Base 3 — Upper bound on s

There is no (118, 241, 830)-net in base 3, because

1 times m-reduction [i] would yield (118, 240, 830)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 433429 589729 276459 326477 824216 577558 954749 737984 291329 882322 222296 319790 825542 053438 870826 984035 330244 979562 013733 > 3²⁴⁰ [i]