MinT - Best Known (77−39, 77, s)-Nets in Base 27

Best Known (77−39, 77, s)-Nets in Base 27

(77−39, 77, 182)-Net over F₂₇ — Constructive and digital

Digital (38, 77, 182)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 28, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]
2. digital (10, 49, 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]

(77−39, 77, 370)-Net in Base 27 — Constructive

(38, 77, 370)-net in base 27, using

11 times m-reduction [i] based on (38, 88, 370)-net in base 27, using
- base change [i] based on digital (16, 66, 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]

(77−39, 77, 475)-Net over F₂₇ — Digital

Digital (38, 77, 475)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−39, 77, 162058)-Net in Base 27 — Upper bound on s

There is no (38, 77, 162059)-net in base 27, because

1 times m-reduction [i] would yield (38, 76, 162059)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 076931 935623 678230 520794 810204 706096 885553 055505 740404 878836 623730 208624 044018 534188 810190 023144 055951 990595 > 27⁷⁶ [i]