MinT - Best Known (68, 68+43, s)-Nets in Base 8

Best Known (68, 68+43, s)-Nets in Base 8

(68, 68+43, 354)-Net over F₈ — Constructive and digital

Digital (68, 111, 354)-net over F₈, using

11 times m-reduction [i] based on digital (68, 122, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 61, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(68, 68+43, 384)-Net in Base 8 — Constructive

(68, 111, 384)-net in base 8, using

t-expansion [i] based on (67, 111, 384)-net in base 8, using
- 1 times m-reduction [i] based on (67, 112, 384)-net in base 8, using
  - trace code for nets [i] based on (11, 56, 192)-net in base 64, using
    - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(68, 68+43, 588)-Net over F₈ — Digital

Digital (68, 111, 588)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+43, 66647)-Net in Base 8 — Upper bound on s

There is no (68, 111, 66648)-net in base 8, because

1 times m-reduction [i] would yield (68, 110, 66648)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2187 878154 336802 995794 236850 914434 265193 756421 336059 681575 692652 755942 189661 682784 937242 276346 906327 > 8¹¹⁰ [i]