MinT - Best Known (69−29, 69, s)-Nets in Base 8

Best Known (69−29, 69, s)-Nets in Base 8

(69−29, 69, 256)-Net over F₈ — Constructive and digital

Digital (40, 69, 256)-net over F₈, using

1 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]

(69−29, 69, 258)-Net in Base 8 — Constructive

(40, 69, 258)-net in base 8, using

1 times m-reduction [i] based on (40, 70, 258)-net in base 8, using
- trace code for nets [i] based on (5, 35, 129)-net in base 64, using
  - base change [i] based on digital (0, 30, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(69−29, 69, 280)-Net over F₈ — Digital

Digital (40, 69, 280)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−29, 69, 21019)-Net in Base 8 — Upper bound on s

There is no (40, 69, 21020)-net in base 8, because

1 times m-reduction [i] would yield (40, 68, 21020)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 720536 272651 126889 128269 055147 416699 193633 480791 469449 375932 > 8⁶⁸ [i]