MinT - Best Known (246−123, 246, s)-Nets in Base 3

Best Known (246−123, 246, s)-Nets in Base 3

(246−123, 246, 78)-Net over F₃ — Constructive and digital

Digital (123, 246, 78)-net over F₃, using

t-expansion [i] based on digital (121, 246, 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]

(246−123, 246, 120)-Net over F₃ — Digital

Digital (123, 246, 120)-net over F₃, using

t-expansion [i] based on digital (113, 246, 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]

(246−123, 246, 912)-Net in Base 3 — Upper bound on s

There is no (123, 246, 913)-net in base 3, because

1 times m-reduction [i] would yield (123, 245, 913)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 802 573317 705356 098283 817546 810652 028047 074919 015682 166943 949951 374096 526331 081885 959392 654363 110662 075595 095110 833459 > 3²⁴⁵ [i]