MinT - Best Known (22, 22+30, s)-Nets in Base 27

Best Known (22, 22+30, s)-Nets in Base 27

(22, 22+30, 116)-Net over F₂₇ — Constructive and digital

Digital (22, 52, 116)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]
2. digital (4, 34, 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]

(22, 22+30, 165)-Net over F₂₇ — Digital

Digital (22, 52, 165)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(22, 22+30, 172)-Net in Base 27 — Constructive

(22, 52, 172)-net in base 27, using

8 times m-reduction [i] based on (22, 60, 172)-net in base 27, using
- base change [i] based on digital (7, 45, 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]

(22, 22+30, 244)-Net in Base 27

(22, 52, 244)-net in base 27, using

base change [i] based on digital (9, 39, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(22, 22+30, 22631)-Net in Base 27 — Upper bound on s

There is no (22, 52, 22632)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 269 851624 767133 865576 735598 249539 462886 363374 544691 585059 731018 984233 821345 > 27⁵² [i]