MinT - Best Known (68−28, 68, s)-Nets in Base 8

Best Known (68−28, 68, s)-Nets in Base 8

(68−28, 68, 256)-Net over F₈ — Constructive and digital

Digital (40, 68, 256)-net over F₈, using

2 times m-reduction [i] based on digital (40, 70, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 35, 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]

(68−28, 68, 300)-Net in Base 8 — Constructive

(40, 68, 300)-net in base 8, using

trace code for nets [i] based on (6, 34, 150)-net in base 64, using
- 1 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
  - base change [i] based on digital (1, 30, 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]

(68−28, 68, 322)-Net over F₈ — Digital

Digital (40, 68, 322)-net over F₈, using

trace code for nets [i] based on digital (6, 34, 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]

(68−28, 68, 21019)-Net in Base 8 — Upper bound on s

There is no (40, 68, 21020)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 720536 272651 126889 128269 055147 416699 193633 480791 469449 375932 > 8⁶⁸ [i]