MinT - Best Known (69, 69+87, s)-Nets in Base 8

Best Known (69, 69+87, s)-Nets in Base 8

(69, 69+87, 113)-Net over F₈ — Constructive and digital

Digital (69, 156, 113)-net over F₈, using

1 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+87, 163)-Net over F₈ — Digital

Digital (69, 156, 163)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+87, 4314)-Net in Base 8 — Upper bound on s

There is no (69, 156, 4315)-net in base 8, because

1 times m-reduction [i] would yield (69, 155, 4315)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 306818 739391 990712 394603 341456 248384 773939 842604 263050 459417 491091 772364 449724 532477 825957 239398 888122 475160 239792 781444 212600 509036 343952 > 8¹⁵⁵ [i]