MinT - Best Known (151−85, 151, s)-Nets in Base 8

Best Known (151−85, 151, s)-Nets in Base 8

(151−85, 151, 111)-Net over F₈ — Constructive and digital

Digital (66, 151, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 52, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (14, 99, 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]

(151−85, 151, 153)-Net over F₈ — Digital

Digital (66, 151, 153)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(151−85, 151, 156)-Net in Base 8

(66, 151, 156)-net in base 8, using

5 times m-reduction [i] based on (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]

(151−85, 151, 3936)-Net in Base 8 — Upper bound on s

There is no (66, 151, 3937)-net in base 8, because

1 times m-reduction [i] would yield (66, 150, 3937)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2920 813819 458411 190028 403796 809053 365874 335147 171282 106468 485827 499019 266550 691746 831297 706994 921335 020047 773274 052131 295561 812515 606432 > 8¹⁵⁰ [i]