MinT - Best Known (66−37, 66, s)-Nets in Base 27

Best Known (66−37, 66, s)-Nets in Base 27

(66−37, 66, 146)-Net over F₂₇ — Constructive and digital

Digital (29, 66, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 22, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (7, 44, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(66−37, 66, 172)-Net in Base 27 — Constructive

(29, 66, 172)-net in base 27, using

22 times m-reduction [i] based on (29, 88, 172)-net in base 27, using
- base change [i] based on digital (7, 66, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(66−37, 66, 230)-Net over F₂₇ — Digital

Digital (29, 66, 230)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−37, 66, 298)-Net in Base 27

(29, 66, 298)-net in base 27, using

2 times m-reduction [i] based on (29, 68, 298)-net in base 27, using
- base change [i] based on digital (12, 51, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(66−37, 66, 42842)-Net in Base 27 — Upper bound on s

There is no (29, 66, 42843)-net in base 27, because

1 times m-reduction [i] would yield (29, 65, 42843)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1093 185780 454645 889921 172106 047229 115537 942950 754659 808021 434041 115500 213712 008137 760205 121029 > 27⁶⁵ [i]