MinT - Best Known (89−45, 89, s)-Nets in Base 8

Best Known (89−45, 89, s)-Nets in Base 8

(89−45, 89, 100)-Net over F₈ — Constructive and digital

Digital (44, 89, 100)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 30, 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, 59, 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]

(89−45, 89, 152)-Net over F₈ — Digital

Digital (44, 89, 152)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−45, 89, 5284)-Net in Base 8 — Upper bound on s

There is no (44, 89, 5285)-net in base 8, because

1 times m-reduction [i] would yield (44, 88, 5285)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 29 716173 027332 124412 708160 313652 471394 661507 553776 819836 806856 138081 553307 814832 > 8⁸⁸ [i]