MinT - Best Known (20, 79, s)-Nets in Base 64

Best Known (20, 79, s)-Nets in Base 64

(20, 79, 177)-Net over F₆₄ — Constructive and digital

Digital (20, 79, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 79, 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]

(20, 79, 257)-Net in Base 64 — Constructive

(20, 79, 257)-net in base 64, using

1 times m-reduction [i] based on (20, 80, 257)-net in base 64, using
- base change [i] based on digital (0, 60, 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]

(20, 79, 342)-Net over F₆₄ — Digital

Digital (20, 79, 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]

(20, 79, 13344)-Net in Base 64 — Upper bound on s

There is no (20, 79, 13345)-net in base 64, because

1 times m-reduction [i] would yield (20, 78, 13345)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 762 452331 370351 593014 381215 423092 192828 303818 138859 418893 263695 048653 256688 286335 008519 944745 578093 332492 863013 389618 561805 757800 478129 919648 > 64⁷⁸ [i]