MinT - Best Known (1, 1+42, s)-Nets in Base 64

Best Known (1, 1+42, s)-Nets in Base 64

(1, 1+42, 80)-Net over F₆₄ — Constructive and digital

Digital (1, 43, 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]

(1, 1+42, 81)-Net over F₆₄ — Digital

Digital (1, 43, 81)-net over F₆₄, using

net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
  - sharp bound on the number of rational points on curves with genus 1 [i]

(1, 1+42, 386)-Net in Base 64 — Upper bound on s

There is no (1, 43, 387)-net in base 64, because

30 times m-reduction [i] would yield (1, 13, 387)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 303643 134427 673040 438880 > 64¹³ [i]