MinT - Best Known (156−90, 156, s)-Nets in Base 8

Best Known (156−90, 156, s)-Nets in Base 8

(156−90, 156, 99)-Net over F₈ — Constructive and digital

Digital (66, 156, 99)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 52, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (14, 104, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(156−90, 156, 144)-Net over F₈ — Digital

Digital (66, 156, 144)-net over F₈, using

t-expansion [i] based on digital (45, 156, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(156−90, 156, 156)-Net in Base 8

(66, 156, 156)-net in base 8, using

base change [i] based on digital (27, 117, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(156−90, 156, 3374)-Net in Base 8 — Upper bound on s

There is no (66, 156, 3375)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 771 676400 669031 015618 784273 217868 663760 585965 911560 360104 082377 660832 657253 477827 872711 829335 467634 938697 156929 722067 302302 291380 026815 175146 > 8¹⁵⁶ [i]