MinT - Best Known (242−127, 242, s)-Nets in Base 3

Best Known (242−127, 242, s)-Nets in Base 3

(242−127, 242, 74)-Net over F₃ — Constructive and digital

Digital (115, 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]

(242−127, 242, 120)-Net over F₃ — Digital

Digital (115, 242, 120)-net over F₃, using

t-expansion [i] based on 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]

(242−127, 242, 751)-Net in Base 3 — Upper bound on s

There is no (115, 242, 752)-net in base 3, because

1 times m-reduction [i] would yield (115, 241, 752)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 753136 063880 269510 653150 065240 398980 298263 307703 061351 369996 118670 332768 543777 947217 077591 432762 519644 266196 298561 > 3²⁴¹ [i]