MinT - Best Known (78−63, 78, s)-Nets in Base 64

Best Known (78−63, 78, s)-Nets in Base 64

(78−63, 78, 177)-Net over F₆₄ — Constructive and digital

Digital (15, 78, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 78, 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]

(78−63, 78, 192)-Net in Base 64 — Constructive

(15, 78, 192)-net in base 64, using

6 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
- base change [i] based on digital (3, 72, 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]

(78−63, 78, 258)-Net over F₆₄ — Digital

Digital (15, 78, 258)-net over F₆₄, using

net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
  - a curve from the manYPoints database [i]

(78−63, 78, 6024)-Net in Base 64 — Upper bound on s

There is no (15, 78, 6025)-net in base 64, because

1 times m-reduction [i] would yield (15, 77, 6025)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11 960782 762863 743587 965905 140694 257683 302830 471137 364059 967554 624310 483941 131231 177835 370048 664274 893703 475989 038822 764100 592173 522061 875584 > 64⁷⁷ [i]