MinT - Best Known (45, 45+49, s)-Nets in Base 27

Best Known (45, 45+49, s)-Nets in Base 27

(45, 45+49, 190)-Net over F₂₇ — Constructive and digital

Digital (45, 94, 190)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 34, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
2. digital (11, 60, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(45, 45+49, 370)-Net in Base 27 — Constructive

(45, 94, 370)-net in base 27, using

t-expansion [i] based on (43, 94, 370)-net in base 27, using
- 14 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(45, 45+49, 460)-Net over F₂₇ — Digital

Digital (45, 94, 460)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+49, 132703)-Net in Base 27 — Upper bound on s

There is no (45, 94, 132704)-net in base 27, because

1 times m-reduction [i] would yield (45, 93, 132704)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 087198 753065 857044 962675 257987 716097 789573 597584 935841 119545 082467 905739 167015 927358 645174 433350 905324 725027 468486 864129 694327 259137 > 27⁹³ [i]