MinT - Best Known (58−23, 58, s)-Nets in Base 8

Best Known (58−23, 58, s)-Nets in Base 8

(58−23, 58, 256)-Net over F₈ — Constructive and digital

Digital (35, 58, 256)-net over F₈, using

2 times m-reduction [i] based on digital (35, 60, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 30, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(58−23, 58, 300)-Net in Base 8 — Constructive

(35, 58, 300)-net in base 8, using

8² times duplication [i] based on (33, 56, 300)-net in base 8, using
- trace code for nets [i] based on (5, 28, 150)-net in base 64, using
  - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(58−23, 58, 322)-Net over F₈ — Digital

Digital (35, 58, 322)-net over F₈, using

trace code for nets [i] based on digital (6, 29, 161)-net over F₆₄, using
- net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
    - a curve from the manYPoints database [i]

(58−23, 58, 33533)-Net in Base 8 — Upper bound on s

There is no (35, 58, 33534)-net in base 8, because

1 times m-reduction [i] would yield (35, 57, 33534)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2993 627046 879444 473249 419179 142336 573056 930843 036334 > 8⁵⁷ [i]