MinT - Best Known (1, 1+5, s)-Nets in Base 64

Best Known (1, 1+5, s)-Nets in Base 64

(1, 1+5, 80)-Net over F₆₄ — Constructive and digital

Digital (1, 6, 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+5, 81)-Net over F₆₄ — Digital

Digital (1, 6, 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+5, 129)-Net in Base 64 — Constructive

(1, 6, 129)-net in base 64, using

1 times m-reduction [i] based on (1, 7, 129)-net in base 64, using
- base change [i] based on digital (0, 6, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(1, 1+5, 735)-Net in Base 64 — Upper bound on s

There is no (1, 6, 736)-net in base 64, because

1 times m-reduction [i] would yield (1, 5, 736)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1076 549041 > 64⁵ [i]