MinT - Best Known (123−65, 123, s)-Nets in Base 8

Best Known (123−65, 123, s)-Nets in Base 8

(123−65, 123, 113)-Net over F₈ — Constructive and digital

Digital (58, 123, 113)-net over F₈, using

1 times m-reduction [i] based on digital (58, 124, 113)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 44, 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, 80, 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]

(123−65, 123, 165)-Net over F₈ — Digital

Digital (58, 123, 165)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−65, 123, 5047)-Net in Base 8 — Upper bound on s

There is no (58, 123, 5048)-net in base 8, because

1 times m-reduction [i] would yield (58, 122, 5048)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 614239 692277 185953 082003 615766 210479 047564 293887 760029 782268 058824 296748 584397 622113 638419 067191 240858 175875 > 8¹²² [i]