MinT - Best Known (247−126, 247, s)-Nets in Base 3

Best Known (247−126, 247, s)-Nets in Base 3

(247−126, 247, 78)-Net over F₃ — Constructive and digital

Digital (121, 247, 78)-net over F₃, using

net from sequence [i] based on digital (121, 77)-sequence over F₃, using
- base reduction for sequences [i] 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]

(247−126, 247, 120)-Net over F₃ — Digital

Digital (121, 247, 120)-net over F₃, using

t-expansion [i] based on digital (113, 247, 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]

(247−126, 247, 841)-Net in Base 3 — Upper bound on s

There is no (121, 247, 842)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7392 501750 019299 619016 664547 786363 557197 068178 140473 338027 010788 404578 435569 379353 491335 781862 889435 307533 263311 245849 > 3²⁴⁷ [i]