MinT - Best Known (242−129, 242, s)-Nets in Base 3

Best Known (242−129, 242, s)-Nets in Base 3

(242−129, 242, 74)-Net over F₃ — Constructive and digital

Digital (113, 242, 74)-net over F₃, using

t-expansion [i] based on digital (107, 242, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(242−129, 242, 120)-Net over F₃ — Digital

Digital (113, 242, 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]

(242−129, 242, 711)-Net in Base 3 — Upper bound on s

There is no (113, 242, 712)-net in base 3, because

1 times m-reduction [i] would yield (113, 241, 712)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 498503 808385 652242 349040 345267 616159 122452 359318 363927 973639 561106 530531 618769 438000 876290 461043 895651 059272 794113 > 3²⁴¹ [i]