MinT - Best Known (42−40, 42, s)-Nets in Base 64

Best Known (42−40, 42, s)-Nets in Base 64

(42−40, 42, 80)-Net over F₆₄ — Constructive and digital

Digital (2, 42, 80)-net over F₆₄, using

t-expansion [i] based on digital (1, 42, 80)-net over F₆₄, using
- net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
    - a function field by SÃ©mirat [i]

(42−40, 42, 97)-Net over F₆₄ — Digital

Digital (2, 42, 97)-net over F₆₄, using

net from sequence [i] based on digital (2, 96)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 2 and N(F) ≥ 97, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(42−40, 42, 671)-Net in Base 64 — Upper bound on s

There is no (2, 42, 672)-net in base 64, because

20 times m-reduction [i] would yield (2, 22, 672)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 5461 266548 447243 503148 198082 588188 881141 > 64²² [i]