MinT - Best Known (248−155, 248, s)-Nets in Base 3

Best Known (248−155, 248, s)-Nets in Base 3

(248−155, 248, 64)-Net over F₃ — Constructive and digital

Digital (93, 248, 64)-net over F₃, using

t-expansion [i] based on digital (89, 248, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(248−155, 248, 96)-Net over F₃ — Digital

Digital (93, 248, 96)-net over F₃, using

t-expansion [i] based on digital (89, 248, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(248−155, 248, 428)-Net in Base 3 — Upper bound on s

There is no (93, 248, 429)-net in base 3, because

1 times m-reduction [i] would yield (93, 247, 429)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8042 242793 223431 201503 858481 548847 123865 869488 971256 833442 664902 286753 153707 266856 431913 455580 474563 271293 425768 403147 > 3²⁴⁷ [i]