MinT - Best Known (19, 19+55, s)-Nets in Base 64

Best Known (19, 19+55, s)-Nets in Base 64

(19, 19+55, 177)-Net over F₆₄ — Constructive and digital

Digital (19, 74, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 74, 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]

(19, 19+55, 257)-Net in Base 64 — Constructive

(19, 74, 257)-net in base 64, using

2 times m-reduction [i] based on (19, 76, 257)-net in base 64, using
- base change [i] based on digital (0, 57, 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]

(19, 19+55, 315)-Net over F₆₄ — Digital

Digital (19, 74, 315)-net over F₆₄, using

net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
  - a curve from the manYPoints database [i]

(19, 19+55, 13243)-Net in Base 64 — Upper bound on s

There is no (19, 74, 13244)-net in base 64, because

1 times m-reduction [i] would yield (19, 73, 13244)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 710576 159599 055093 841563 243804 443831 603739 096209 214481 257525 131757 195970 430495 077633 445411 547098 128373 156265 782737 021761 721813 697405 > 64⁷³ [i]