MinT - Best Known (69, 69+85, s)-Nets in Base 8

Best Known (69, 69+85, s)-Nets in Base 8

(69, 69+85, 113)-Net over F₈ — Constructive and digital

Digital (69, 154, 113)-net over F₈, using

3 times m-reduction [i] based on digital (69, 157, 113)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 55, 48)-net over F₈, using
    - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
        a function field by SÃ©mirat [i]
  2. digital (14, 102, 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]

(69, 69+85, 168)-Net over F₈ — Digital

Digital (69, 154, 168)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+85, 4571)-Net in Base 8 — Upper bound on s

There is no (69, 154, 4572)-net in base 8, because

1 times m-reduction [i] would yield (69, 153, 4572)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 500152 147931 807776 978962 385502 122124 111249 067607 824111 136376 622623 278112 903173 135610 824004 696611 694602 381985 077400 778576 863638 921150 492382 > 8¹⁵³ [i]