MinT - Best Known (167−97, 167, s)-Nets in Base 8

Best Known (167−97, 167, s)-Nets in Base 8

(167−97, 167, 100)-Net over F₈ — Constructive and digital

Digital (70, 167, 100)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 56, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [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, 111, 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]

(167−97, 167, 147)-Net over F₈ — Digital

Digital (70, 167, 147)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−97, 167, 156)-Net in Base 8

(70, 167, 156)-net in base 8, using

5 times m-reduction [i] based on (70, 172, 156)-net in base 8, using
- base change [i] based on digital (27, 129, 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]

(167−97, 167, 3525)-Net in Base 8 — Upper bound on s

There is no (70, 167, 3526)-net in base 8, because

1 times m-reduction [i] would yield (70, 166, 3526)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 829076 774136 074624 669360 289779 506829 321713 558935 951047 979867 564253 907362 451763 693124 134414 936573 795715 893964 414019 441676 620919 512635 752189 416894 867940 > 8¹⁶⁶ [i]