MinT - Best Known (123, 123+127, s)-Nets in Base 3

Best Known (123, 123+127, s)-Nets in Base 3

(123, 123+127, 78)-Net over F₃ — Constructive and digital

Digital (123, 250, 78)-net over F₃, using

t-expansion [i] based on digital (121, 250, 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]

(123, 123+127, 120)-Net over F₃ — Digital

Digital (123, 250, 120)-net over F₃, using

t-expansion [i] based on digital (113, 250, 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]

(123, 123+127, 873)-Net in Base 3 — Upper bound on s

There is no (123, 250, 874)-net in base 3, because

1 times m-reduction [i] would yield (123, 249, 874)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66607 357068 439437 657713 672506 746136 736916 264379 892734 925026 594665 924232 453606 318677 221344 181595 125580 058240 738491 721625 > 3²⁴⁹ [i]