MinT - Best Known (18−16, 18, s)-Nets in Base 64

Best Known (18−16, 18, s)-Nets in Base 64

(18−16, 18, 80)-Net over F₆₄ — Constructive and digital

Digital (2, 18, 80)-net over F₆₄, using

t-expansion [i] based on digital (1, 18, 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]

(18−16, 18, 97)-Net over F₆₄ — Digital

Digital (2, 18, 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]

(18−16, 18, 688)-Net in Base 64 — Upper bound on s

There is no (2, 18, 689)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 325 969419 386743 933113 467802 919090 > 64¹⁸ [i]