MinT - Best Known (21, 21+67, s)-Nets in Base 64

Best Known (21, 21+67, s)-Nets in Base 64

(21, 21+67, 177)-Net over F₆₄ — Constructive and digital

Digital (21, 88, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 88, 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]

(21, 21+67, 216)-Net in Base 64 — Constructive

(21, 88, 216)-net in base 64, using

t-expansion [i] based on (18, 88, 216)-net in base 64, using
- 3 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
  - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(21, 21+67, 342)-Net over F₆₄ — Digital

Digital (21, 88, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 88, 342)-net over F₆₄, using
- net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]

(21, 21+67, 12054)-Net in Base 64 — Upper bound on s

There is no (21, 88, 12055)-net in base 64, because

1 times m-reduction [i] would yield (21, 87, 12055)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 13 730814 344109 969567 189109 181486 268473 664443 257781 136645 492523 878336 067833 650078 616434 007222 520922 115304 244057 969216 812061 206617 281128 388795 173502 051437 565680 > 64⁸⁷ [i]