MinT - Best Known (66, 66+39, s)-Nets in Base 8

Best Known (66, 66+39, s)-Nets in Base 8

(66, 66+39, 354)-Net over F₈ — Constructive and digital

Digital (66, 105, 354)-net over F₈, using

13 times m-reduction [i] based on digital (66, 118, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 59, 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]

(66, 66+39, 514)-Net in Base 8 — Constructive

(66, 105, 514)-net in base 8, using

8¹ times duplication [i] based on (65, 104, 514)-net in base 8, using
- base change [i] based on digital (39, 78, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 39, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(66, 66+39, 691)-Net over F₈ — Digital

Digital (66, 105, 691)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+39, 99379)-Net in Base 8 — Upper bound on s

There is no (66, 105, 99380)-net in base 8, because

1 times m-reduction [i] would yield (66, 104, 99380)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8344 907892 897408 011075 669544 622820 218486 658345 384113 057396 123379 910049 854843 432234 630577 016577 > 8¹⁰⁴ [i]